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<title>Weierstraßsche ℘-Funktion</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Weierstraßsche ℘-Funktion</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">In der Mathematik bezeichnet die Weierstraßsche ℘-Funktion (sprich „… p-Funktion“, siehe <a href="Weierstra%C3%9F-p" title="Weierstraß-p">Weierstraß-p</a>) eine bestimmte <a href="Elliptische_Funktion" title="Elliptische Funktion">elliptische Funktion</a> in Abhängigkeit eines <a href="Gitter_(Mathematik)" title="Gitter (Mathematik)">Gitters</a>. Benannt ist sie nach dem Mathematiker <a href="Karl_Weierstra%C3%9F" title="Karl Weierstraß">Karl Weierstraß</a>. Mithilfe der Weierstraßschen ℘-Funktion und ihrer Ableitung lassen sich <a href="Elliptische_Kurven" class="mw-redirect" title="Elliptische Kurven">elliptische Kurven</a> über den <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a> parametrisieren.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} \setminus \{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} \setminus \{0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36fe48a3fdfcfc5fab98a97ef3607e48931fb2b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} \setminus \{0\}}" loading="lazy"> zwei komplexe Zahlen, welche über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear unabhängig</a> sind und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3017b5b8e08ec4ff81bf287eb5c2cdbaba267be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.756ex; height:2.843ex;" alt="{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}" loading="lazy"> das <a href="Gitter_(Mathematik)" title="Gitter (Mathematik)">Gitter</a>, das von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b914a8bfef5d1b9b106048afa0aab4a99251f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}" loading="lazy"> erzeugt wird. Dann ist die ℘-Funktion zum Gitter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> wie folgt definiert:

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<annotation encoding="application/x-tex">{\displaystyle \wp (z,\omega _{1},\omega _{2}):=\wp (z,\Lambda ):={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0418f6d0eacf1cb93be526861c89f215b05fed9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:57.139ex; height:7.176ex;" alt="{\displaystyle \wp (z,\omega _{1},\omega _{2}):=\wp (z,\Lambda ):={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)}" loading="lazy"></dd></dl>

Die Reihe konvergiert <a href="Lokal_gleichm%C3%A4%C3%9Fige_Konvergenz" title="Lokal gleichmäßige Konvergenz">lokal gleichmäßig</a> absolut in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \setminus \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \setminus \Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a55b01c4e41551e4496e6f67bf573fa747869ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.486ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \setminus \Lambda }" loading="lazy">. Häufig wird statt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (z,\omega _{1},\omega _{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \wp (z,\omega _{1},\omega _{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8ea87ec5f0916812f3fdc32c4721a27045b3f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.444ex; height:2.843ex;" alt="{\displaystyle \wp (z,\omega _{1},\omega _{2})}" loading="lazy"> auch nur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (z)}">

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<annotation encoding="application/x-tex">{\displaystyle \wp (z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31c887ae905d692694de7e982ac1d6ad56e1f3b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.376ex; height:2.843ex;" alt="{\displaystyle \wp (z)}" loading="lazy"> geschrieben.

Die Weierstraßsche ℘-Funktion ist gerade so konstruiert, dass sie einen Pol der Ordnung 2 an jeder Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda \in \Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1f14e3b08f79e38d62384eb90eff43a2fb587c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.809ex; height:2.176ex;" alt="{\displaystyle \lambda \in \Lambda }" loading="lazy"> hat. Da die Summe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e58b477b0b6282bc45e094ccbbda98e6d66a47be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.726ex; height:6.509ex;" alt="{\displaystyle \sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{2}}}}" loading="lazy"> alleine nicht absolut konvergieren würde, ist es nötig, den Term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1}{\lambda ^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{\lambda ^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/093a89232224dd21be4d5a6c454226bd84c4b761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:5.054ex; height:5.509ex;" alt="{\displaystyle -{\frac {1}{\lambda ^{2}}}}" loading="lazy"> hinzuzufügen.<a href="#cite_note-:0-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

Eine Kubik der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4430cfee774eeb483752672c90dd0fc2043d637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.042ex; height:3.343ex;" alt="{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2},g_{3}\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle g_{2},g_{3}\in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e2a055c7778e8dc686739e4f1a416eb153a2fac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.879ex; height:2.509ex;" alt="{\displaystyle g_{2},g_{3}\in \mathbb {C} }" loading="lazy"> komplexe Zahlen sind mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90947cb570ae5f036be4b6fc2c13074df4167da5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.771ex; height:3.176ex;" alt="{\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0}" loading="lazy">, lässt sich nicht rational parametrisieren.<a href="#cite_note-:5-2">[2]</a> Dennoch würde man gerne eine Parametrisierung finden.

Für die <a href="Quadrik" title="Quadrik">Quadrik</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\}}">

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<annotation encoding="application/x-tex">{\displaystyle K=\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31b58ffe3912f3a6480beefc2292bdd440db51e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.028ex; height:3.176ex;" alt="{\displaystyle K=\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\}}" loading="lazy">, also den Einheitskreis, existiert bekanntlich eine (nichtrationale) Parametrisierung durch die <a href="Sinus_und_Kosinus" title="Sinus und Kosinus">Sinusfunktion</a> und deren Ableitung, die Kosinusfunktion:

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<annotation encoding="application/x-tex">{\displaystyle \psi :\mathbb {R} /2\pi \mathbb {Z} \to K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a05ace26dec8893f35a922db2582a5719b9a16cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.016ex; height:2.843ex;" alt="{\displaystyle \psi :\mathbb {R} /2\pi \mathbb {Z} \to K}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto (\sin(t),\cos(t))}">

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<annotation encoding="application/x-tex">{\displaystyle t\mapsto (\sin(t),\cos(t))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19dcadde9908092e5957082c8b5c7c01b2051679.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.561ex; height:2.843ex;" alt="{\displaystyle t\mapsto (\sin(t),\cos(t))}" loading="lazy">.</dd></dl>

Wegen der Periodizität des Sinus und des Kosinus ist hier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} /2\pi \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} /2\pi \mathbb {Z} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd6ed3ed42d3ae16ae67a7049e8bad73dfb00e14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.885ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} /2\pi \mathbb {Z} }" loading="lazy"> als Definitionsbereich gewählt, um eine injektive Abbildung zu erhalten.

Auf ganz analoge Weise erhält man auch eine Parametrisierung der Kubik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bf11c7b76df96817b8d361f235e41d736329d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.583ex; height:3.176ex;" alt="{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }}" loading="lazy"> mit der doppeltperiodischen ℘-Funktion (siehe im Abschnitt „Zusammenhang mit elliptischen Kurven“). Diese Parametrisierung hat dann den Definitionsbereich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} /\Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} /\Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9326cf3ac2c0ad37d1d33f02c3bbb927e0af65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.454ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} /\Lambda }" loading="lazy">, was topologisch einem <a href="Torus" title="Torus">Torus</a> entspricht.<a href="#cite_note-3">[3]</a>

Es gibt noch eine weitere Analogie zu den trigonometrischen Funktionen. Betrachtet man die Integralfunktion

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<annotation encoding="application/x-tex">{\displaystyle a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {(1-y^{2})}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9246685285982300a5c6f9c74e3e639e6422915b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.678ex; height:6.676ex;" alt="{\displaystyle a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {(1-y^{2})}}}}" loading="lazy">,</dd></dl>

dann lässt sich diese durch die Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\sin(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y=\sin(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/271d3e185f727418bba060e3c26c9a4b012c7b8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.758ex; height:2.843ex;" alt="{\displaystyle y=\sin(t)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\arcsin(x)}">

<mi>arcsin</mi>

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<annotation encoding="application/x-tex">{\displaystyle s=\arcsin(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/deb6faf45e442fa51fd56f51e95ad36729022819.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.29ex; height:2.843ex;" alt="{\displaystyle s=\arcsin(x)}" loading="lazy"> vereinfachen. Dadurch ergibt sich:

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<mi>arcsin</mi>

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<annotation encoding="application/x-tex">{\displaystyle a(x)=\int _{0}^{s}dt=s=\arcsin(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8d983c777453a39b3139018124d532e97c29ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.766ex; height:5.843ex;" alt="{\displaystyle a(x)=\int _{0}^{s}dt=s=\arcsin(x)}" loading="lazy"></dd></dl>

Das bedeutet, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}(x)=\sin(x)}">

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<annotation encoding="application/x-tex">{\displaystyle a^{-1}(x)=\sin(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a7671fde98a86064b6bb59f9da72c63f580e549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.795ex; height:3.176ex;" alt="{\displaystyle a^{-1}(x)=\sin(x)}" loading="lazy">. Also erhält man den Sinus als <a href="Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktion</a> einer Integralfunktion.<a href="#cite_note-4">[4]</a>

Auch elliptische Funktionen sind Umkehrfunktionen von Integralfunktionen, den <a href="Elliptisches_Integral" class="mw-redirect" title="Elliptisches Integral">elliptischen Integralen</a>. Insbesondere erhält man die ℘-Funktion auf folgende Weise:

Sei

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(z)=-\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/117b43f1ecbb35a6c9d6da22edcdba19a244bf73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.812ex; height:6.676ex;" alt="{\displaystyle u(z)=-\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}}" loading="lazy">.</dd></dl>

Dann lässt sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4bafc86c3426745a84a6074675a2f8bbee9ed76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.662ex; height:2.676ex;" alt="{\displaystyle u^{-1}}" loading="lazy"> auf die komplexe Ebene fortsetzen und entspricht der ℘-Funktion.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy"> ist eine <a href="Meromorphe_Funktion" title="Meromorphe Funktion">meromorphe Funktion</a> mit einem Pol der Ordnung 2 an jedem Gitterpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \in \Lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1f14e3b08f79e38d62384eb90eff43a2fb587c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.809ex; height:2.176ex;" alt="{\displaystyle \lambda \in \Lambda }" loading="lazy">.<a href="#cite_note-6">[6]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy"> ist eine <a href="Homogene_Funktion" title="Homogene Funktion">homogene Funktion</a>, denn es gilt</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp (\lambda z,\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-2}\wp (z,\omega _{1},\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a2b7e391d9388262fff3bb9feed44e7449a1dd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.74ex; height:3.176ex;" alt="{\displaystyle \wp (\lambda z,\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-2}\wp (z,\omega _{1},\omega _{2})}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/169fae60c23a2027ece2aa7fd4b5047492887e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {C} \setminus \{0\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {C} \setminus \{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d989c233654e602d28ce0f2ecf4088fbef5ec9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.556ex; height:2.843ex;" alt="{\displaystyle \lambda \in \mathbb {C} \setminus \{0\}}" loading="lazy">.</dd></dl>

<ul><li>℘ ist eine gerade Funktion. Das heißt, es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (z)=\wp (-z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp (z)=\wp (-z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab3e18f1a7d183c36cd10c1fa44e3d974e30f9e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.658ex; height:2.843ex;" alt="{\displaystyle \wp (z)=\wp (-z)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} \setminus \Lambda }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} \setminus \Lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a97129f9b2401d1639edab9125d44eb8846c16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.415ex; height:2.843ex;" alt="{\displaystyle z\in \mathbb {C} \setminus \Lambda }" loading="lazy">, wie man auf folgende Weise sieht:</li></ul>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\wp (-z)&={\frac {1}{(-z)^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(-z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)=\wp (z)\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0017bbe7f16448cdbbdfae84dc36b115d2606c15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.138ex; margin-bottom: -0.2ex; width:48.69ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}\wp (-z)&={\frac {1}{(-z)^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(-z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)=\wp (z)\end{aligned}}}" loading="lazy"></dd></dl>

Die vorletzte Gleichheit folgt aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-\lambda :\lambda \in \Lambda \}=\Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{-\lambda :\lambda \in \Lambda \}=\Lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c67b0e7647c9f6601574c871bc9cf9b3aa2ec5c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.946ex; height:2.843ex;" alt="{\displaystyle \{-\lambda :\lambda \in \Lambda \}=\Lambda }" loading="lazy">. Da die Summe absolut konvergiert, ändert diese Umordnung am Grenzwert nichts.

<ul><li>Die Ableitung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy"> ist gegeben durch</li></ul>

<msup>

</msup>

</mrow>

</munder>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/757b30ea7671c91186b613d8144554b057dccf2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.242ex; height:6.509ex;" alt="{\displaystyle \wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}}" loading="lazy">.<a href="#cite_note-:1-7">[7]</a></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp '}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp '}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98cc7b9ef41a255d70915227b703405b168cd8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.163ex; height:3.009ex;" alt="{\displaystyle \wp '}" loading="lazy"> sind doppeltperiodisch mit den Perioden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b914a8bfef5d1b9b106048afa0aab4a99251f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}" loading="lazy">. Das bedeutet, es gilt<a href="#cite_note-:1-7">[7]</a>:</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp (z+\omega _{1})=\wp (z)=\wp (z+\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dc4dd23c766962beb5dd898f4658b6e391a4279.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.005ex; height:2.843ex;" alt="{\displaystyle \wp (z+\omega _{1})=\wp (z)=\wp (z+\omega _{2})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp '(z+\omega _{1})=\wp '(z)=\wp '(z+\omega _{2})}">

<msup>

</msup>

<msub>

</mrow>

</msub>

<msup>

</msup>

<msup>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp '(z+\omega _{1})=\wp '(z)=\wp '(z+\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d53672edd60be232bf77b5e9bc06ad619fb1a9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.059ex; height:3.009ex;" alt="{\displaystyle \wp '(z+\omega _{1})=\wp '(z)=\wp '(z+\omega _{2})}" loading="lazy">.</dd></dl>

Daraus folgt, dass für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \in \Lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d1f14e3b08f79e38d62384eb90eff43a2fb587c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.809ex; height:2.176ex;" alt="{\displaystyle \lambda \in \Lambda }" loading="lazy"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (z+\lambda )=\wp (z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp (z+\lambda )=\wp (z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e9c5ad654e67c41b7d7231f78b25469dc0420a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.046ex; height:2.843ex;" alt="{\displaystyle \wp (z+\lambda )=\wp (z)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp '(z+\lambda )=\wp '(z)}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp '(z+\lambda )=\wp '(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cccef62e35e4170c9e54411c42ebdfdb3744a61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.415ex; height:3.009ex;" alt="{\displaystyle \wp '(z+\lambda )=\wp '(z)}" loading="lazy">. Funktionen, die meromorph und doppeltperiodisch sind, nennt man auch <a href="Elliptische_Funktion" title="Elliptische Funktion">elliptische Funktionen</a>.

<div class="mw-heading mw-heading2"><h2 id="Laurent-Entwicklung">Laurent-Entwicklung</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:=\min\{{|\omega }|:\omega \neq 0\}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r:=\min\{{|\omega }|:\omega \neq 0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88672a4ad82158b55d7409679b9d4c744810d4e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.377ex; height:2.843ex;" alt="{\displaystyle r:=\min\{{|\omega }|:\omega \neq 0\}}" loading="lazy">. Dann hat die ℘-Funktion für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<|z|<r}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<|z|<r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f5478036c4c88295942b47572da20bd54e7c32b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.79ex; height:2.843ex;" alt="{\displaystyle 0<|z|<r}" loading="lazy"> folgende <a href="Laurent-Reihe" title="Laurent-Reihe">Laurent-Reihe</a>:

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }(2n+1)G_{2n+2}z^{2n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc7987f22e692dd67a16ba3d28210fb0388683a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.118ex; height:6.843ex;" alt="{\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }(2n+1)G_{2n+2}z^{2n}}" loading="lazy">,</dd></dl>

wobei

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{n}=\sum _{0\neq \omega \in \Lambda }\omega ^{-n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee3331ed58b5aee35e74fae8e7ef6b063d3ff0ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.833ex; height:6.009ex;" alt="{\displaystyle G_{n}=\sum _{0\neq \omega \in \Lambda }\omega ^{-n}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73136e4a27fe39c123d16a7808e76d3162ce42bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}" loading="lazy"> sogenannte <a href="Eisensteinreihe" title="Eisensteinreihe">Eisensteinreihen</a> sind.<a href="#cite_note-:1-7">[7]</a></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Differentialgleichung">Differentialgleichung</h2></div>

Wir setzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}=60G_{4}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}=60G_{4}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62a2dcf9e1c8db0b623528d10511d21469c67e82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle g_{2}=60G_{4}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}=140G_{6}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{3}=140G_{6}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b19924e226a49ed082e86d3779360050bf8d6301.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.63ex; height:2.509ex;" alt="{\displaystyle g_{3}=140G_{6}}" loading="lazy">. Dann erfüllt die ℘-Funktion folgende <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichung</a><a href="#cite_note-:1-7">[7]</a>:

<msup>

<mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5e951985edd46e52c8e1b3de72b11987cb7f880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.956ex; height:3.176ex;" alt="{\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}}" loading="lazy">.</dd></dl>

Dies lässt sich verifizieren, indem man den Pol an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b92bfc06485cc90286474b14a516a68d8bfdd7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}" loading="lazy"> durch eine Linearkombination von Potenzen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp '}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp '}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98cc7b9ef41a255d70915227b703405b168cd8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.163ex; height:3.009ex;" alt="{\displaystyle \wp '}" loading="lazy"> eliminiert. Dann erhält man eine ganze, elliptische Funktion, die nach dem <a href="Satz_von_Liouville_(Funktionentheorie)" title="Satz von Liouville (Funktionentheorie)">Satz von Liouville</a> konstant sein muss.

<div class="mw-heading mw-heading2"><h2 id="Invarianten_und_modulare_Diskriminante">Invarianten und modulare Diskriminante</h2></div>

Die Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207852a9b2705498c65b929b79fdda27b257ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}" loading="lazy">, die in der Differentialgleichung auftauchen, heißen die Invarianten. Man betrachtet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207852a9b2705498c65b929b79fdda27b257ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}" loading="lazy"> als Funktionen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b914a8bfef5d1b9b106048afa0aab4a99251f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}" loading="lazy"> und definiert die Diskriminante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta :=g_{2}^{3}-27g_{3}^{2}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta :=g_{2}^{3}-27g_{3}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/070ab30d5132c828e99b0a7086955deb4991d875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.191ex; height:3.176ex;" alt="{\displaystyle \Delta :=g_{2}^{3}-27g_{3}^{2}}" loading="lazy">.

Wie man an der Eisensteinreihe erkennen kann, sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207852a9b2705498c65b929b79fdda27b257ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}" loading="lazy"> <a href="Homogene_Funktion" title="Homogene Funktion">homogene</a> Funktionen vom Grad −4 und −6. Das heißt, es gilt:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34e14da18e5ff61ee4d66c60022122ec3cc2623f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.51ex; height:3.176ex;" alt="{\displaystyle g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58b279909492e58b455ab143d4f6cfba3ea9499c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.51ex; height:3.176ex;" alt="{\displaystyle g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-12}\Delta (\omega _{1},\omega _{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta (\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-12}\Delta (\omega _{1},\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b124cfc3f148b519d44ff56578b3e42e269e9861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.877ex; height:3.176ex;" alt="{\displaystyle \Delta (\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-12}\Delta (\omega _{1},\omega _{2})}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda \neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dedcbbf652c56a75cb890f36b4ff1f3bd984af45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.676ex;" alt="{\displaystyle \lambda \neq 0}" loading="lazy">.<a href="#cite_note-:2-8">[8]</a></dd></dl>

Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b914a8bfef5d1b9b106048afa0aab4a99251f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}" loading="lazy"> so gewählt sind, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} \left({\frac {\omega _{2}}{\omega _{1}}}\right)>0}">

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} \left({\frac {\omega _{2}}{\omega _{1}}}\right)>0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f30399aa5910bca0bfa053e5eeda334fac821c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.794ex; height:6.176ex;" alt="{\displaystyle \operatorname {Im} \left({\frac {\omega _{2}}{\omega _{1}}}\right)>0}" loading="lazy">, können <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2},g_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2},g_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b93bd08c229189afed60922347240293f9c903b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.36ex; height:2.009ex;" alt="{\displaystyle g_{2},g_{3}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> als Funktionen in einer komplexen Variablen in der oberen Halbebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c57a494beaf8b794ea4125129a756ec64b8c6363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.356ex; height:2.843ex;" alt="{\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}" loading="lazy"> aufgefasst werden.

Dazu setzt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aae508e0d16371bbee7c6014aa84b19c71060381.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.637ex; height:5.009ex;" alt="{\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}" loading="lazy"> und erhält:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec8d1be559609664697e2aa5ff8132a5a6e4bb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.976ex; height:3.176ex;" alt="{\displaystyle g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2})}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afb8fc26fad84ac0d26c4c2a92a59b5dc2a12e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.976ex; height:3.176ex;" alt="{\displaystyle g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (1,\tau )=\omega _{1}^{12}\Delta (\omega _{1},\omega _{2})}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta (1,\tau )=\omega _{1}^{12}\Delta (\omega _{1},\omega _{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fade083fa3e880a3f0461f5edbb9868a1ac85ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.343ex; height:3.176ex;" alt="{\displaystyle \Delta (1,\tau )=\omega _{1}^{12}\Delta (\omega _{1},\omega _{2})}" loading="lazy">.<a href="#cite_note-:2-8">[8]</a></dd></dl>

Also werden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207852a9b2705498c65b929b79fdda27b257ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

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<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> dadurch nur skaliert. Man setzt nun:

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<annotation encoding="application/x-tex">{\displaystyle g_{2}(\tau ):=g_{2}(1,\tau )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d37e2d4ac876c6e86ece998b170687267c95a40d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.291ex; height:2.843ex;" alt="{\displaystyle g_{2}(\tau ):=g_{2}(1,\tau )}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}(\tau ):=g_{3}(1,\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle g_{3}(\tau ):=g_{3}(1,\tau )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd144bcddcd540a50ce361346b031639823e73b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.291ex; height:2.843ex;" alt="{\displaystyle g_{3}(\tau ):=g_{3}(1,\tau )}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (\tau ):=\Delta (1,\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta (\tau ):=\Delta (1,\tau )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5cf48a71a8fd90cdbaca9e37f10319bdc9786fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.836ex; height:2.843ex;" alt="{\displaystyle \Delta (\tau ):=\Delta (1,\tau )}" loading="lazy"></dd></dl>

Damit erhält man sogenannte <a href="Modulform" class="mw-redirect" title="Modulform">Modulformen</a>. Auch die ℘-Funktion kann auf diese Weise als Modulform aufgefasst werden.

<div class="mw-heading mw-heading2"><h2 id="Die_Konstanten_e1,_e2_und_e3">Die Konstanten e1, e2 und e3</h2></div>

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e81caf3d4bcb929315801cbabc83543829484ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4045b5c7cee9bd0681153bbb077489b13269355e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1686693c185b6df4a5f57f34f7b61c4a0e057ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{3}}" loading="lazy"> werden die folgendermaßen definierten Halbwerte der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp }">

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<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy">-Funktion bezeichnet.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e_{1}&:=\wp \left({\frac {\omega _{1}}{2}}\right)\\e_{2}&:=\wp \left({\frac {\omega _{2}}{2}}\right)\\e_{3}&:=\wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b9b90c13029b774024e64c504551c5f8d9e7a54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:20.598ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}e_{1}&:=\wp \left({\frac {\omega _{1}}{2}}\right)\\e_{2}&:=\wp \left({\frac {\omega _{2}}{2}}\right)\\e_{3}&:=\wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)\end{aligned}}}" loading="lazy"></dd></dl>

Sie sind <a href="Paarweise_verschieden" title="Paarweise verschieden">paarweise verschieden</a> und hängen vom Gitter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> ab, aber nicht von der Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1},\omega _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{1},\omega _{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc9e7a34c1cd21e1fda694fca931aab81277773b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.034ex; height:2.009ex;" alt="{\displaystyle \omega _{1},\omega _{2}}" loading="lazy">.<a href="#cite_note-9">[9]</a>

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<annotation encoding="application/x-tex">{\displaystyle e_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e81caf3d4bcb929315801cbabc83543829484ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4045b5c7cee9bd0681153bbb077489b13269355e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1686693c185b6df4a5f57f34f7b61c4a0e057ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{3}}" loading="lazy"> sind Nullstellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle 4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0301d2ef4f8c3f066dc935b624139da1462d452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.976ex; height:3.176ex;" alt="{\displaystyle 4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}" loading="lazy"> und erfüllen die Gleichung

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<annotation encoding="application/x-tex">{\displaystyle e_{1}+e_{2}+e_{3}=0.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1eff1a740014668723501af8c452c38c4609d17a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.002ex; height:2.509ex;" alt="{\displaystyle e_{1}+e_{2}+e_{3}=0.}" loading="lazy"></dd></dl>

Weil diese Nullstellen verschieden sind, verschwindet die Diskriminante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

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<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> nicht in der oberen Halbebene.<a href="#cite_note-10">[10]</a>

Die Differentialgleichung lässt sich in der folgenden Form schreiben:

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<annotation encoding="application/x-tex">{\displaystyle \wp '^{2}(z)=4\,(\wp (z)-e_{1})\,(\wp (z)-e_{2})\,(\wp (z)-e_{3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d559b31b8e98d1e1e0347986cb0409f02c6fcf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.794ex; height:3.176ex;" alt="{\displaystyle \wp '^{2}(z)=4\,(\wp (z)-e_{1})\,(\wp (z)-e_{2})\,(\wp (z)-e_{3})}" loading="lazy"></dd></dl>

Dies bedeutet, dass die Halbperioden Nullstellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp '}">

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<annotation encoding="application/x-tex">{\displaystyle \wp '}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98cc7b9ef41a255d70915227b703405b168cd8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.163ex; height:3.009ex;" alt="{\displaystyle \wp '}" loading="lazy"> sind.

Die Invarianten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle g_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207852a9b2705498c65b929b79fdda27b257ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{3}}" loading="lazy"> lassen sich auf folgende Weise durch die Konstanten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},e_{2},e_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{1},e_{2},e_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e35821963b9761d80f5b4079f1ba179ac101da80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.481ex; height:2.009ex;" alt="{\displaystyle e_{1},e_{2},e_{3}}" loading="lazy"> ausdrücken:<a href="#cite_note-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{2}&=-4\,(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})\\g_{3}&=4\,e_{1}e_{2}e_{3}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ccb42c75b96bab0d2713ad6ebf385566b9f59fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.687ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}g_{2}&=-4\,(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})\\g_{3}&=4\,e_{1}e_{2}e_{3}\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle e_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e81caf3d4bcb929315801cbabc83543829484ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4045b5c7cee9bd0681153bbb077489b13269355e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1686693c185b6df4a5f57f34f7b61c4a0e057ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{3}}" loading="lazy"> hängen mit der <a href="Elliptische_Lambda-Funktion" title="Elliptische Lambda-Funktion">elliptischen Lambda-Funktion</a> zusammen:

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<annotation encoding="application/x-tex">{\displaystyle \lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd95bee64a70a4a8bd349b935abfab5d23559261.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.057ex; height:5.343ex;" alt="{\displaystyle \lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beziehung_zu_den_jacobischen_elliptischen_Funktionen">Beziehung zu den jacobischen elliptischen Funktionen</h2></div>

Für numerische Zwecke ist es oft zweckmäßig, die weierstraßsche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp }">

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<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy">-Funktion durch die <a href="Jacobische_elliptische_Funktionen" title="Jacobische elliptische Funktionen">jacobischen elliptischen Funktionen</a> auszudrücken.

Die grundlegenden Beziehungen dafür sind:<a href="#cite_note-12">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle \wp (z)=e_{3}+{\frac {e_{1}-e_{3}}{\operatorname {sn} ^{2}w}}=e_{2}+(e_{1}-e_{3}){\frac {\operatorname {dn} ^{2}w}{\operatorname {sn} ^{2}w}}=e_{1}+(e_{1}-e_{3}){\frac {\operatorname {cn} ^{2}w}{\operatorname {sn} ^{2}w}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9dd55cbcb43359a33d5f4c3328226fb7d2335a0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:67.848ex; height:6.009ex;" alt="{\displaystyle \wp (z)=e_{3}+{\frac {e_{1}-e_{3}}{\operatorname {sn} ^{2}w}}=e_{2}+(e_{1}-e_{3}){\frac {\operatorname {dn} ^{2}w}{\operatorname {sn} ^{2}w}}=e_{1}+(e_{1}-e_{3}){\frac {\operatorname {cn} ^{2}w}{\operatorname {sn} ^{2}w}}.}" loading="lazy"></dd></dl>

Dabei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e81caf3d4bcb929315801cbabc83543829484ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4045b5c7cee9bd0681153bbb077489b13269355e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{2}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1686693c185b6df4a5f57f34f7b61c4a0e057ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{3}}" loading="lazy"> die oben beschriebenen Halbwerte. Das Argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

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<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> ist gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle w=z{\sqrt {e_{1}-e_{3}}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0807e6b8caae7e28f4d0f4964909177d43af3a62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.549ex; height:3.009ex;" alt="{\displaystyle w=z{\sqrt {e_{1}-e_{3}}}.}" loading="lazy"></dd></dl>

Für den elliptischen Modul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> der Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sn} ,\operatorname {cn} ,\operatorname {dn} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sn} ,\operatorname {cn} ,\operatorname {dn} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9f21adde5bd13d6e5fb3f9d9f38e46f387a853a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.187ex; height:2.509ex;" alt="{\displaystyle \operatorname {sn} ,\operatorname {cn} ,\operatorname {dn} }" loading="lazy"> gilt

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<annotation encoding="application/x-tex">{\displaystyle k={\sqrt {\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29d5189277327faac2000042ebf5cbaa7dd82468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.232ex; height:6.343ex;" alt="{\displaystyle k={\sqrt {\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beziehung_zu_den_jacobischen_Thetafunktionen">Beziehung zu den jacobischen Thetafunktionen</h2></div>

Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp }">

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<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy"> lässt sich auch durch die <a href="Jacobische_Thetafunktion" title="Jacobische Thetafunktion">jacobischen Thetafunktionen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{1},\vartheta _{2},\vartheta _{3},\vartheta _{4}}">

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<annotation encoding="application/x-tex">{\displaystyle \vartheta _{1},\vartheta _{2},\vartheta _{3},\vartheta _{4}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdfa550ed109a628c21b7d2397902426aad5f54f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.814ex; height:2.509ex;" alt="{\displaystyle \vartheta _{1},\vartheta _{2},\vartheta _{3},\vartheta _{4}}" loading="lazy"> ausdrücken:<a href="#cite_note-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle \wp (z,\omega _{1},\omega _{2})=e_{1}+\left({\frac {\pi \vartheta _{3}(0,\tau )\vartheta _{4}(0,\tau )\vartheta _{2}(\pi z/\omega _{1},\tau )}{\omega _{1}\vartheta _{1}(\pi z/\omega _{1},\tau )}}\right)^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35ff22956b70f1aa45d7ec870719918c441bb645.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.991ex; height:6.843ex;" alt="{\displaystyle \wp (z,\omega _{1},\omega _{2})=e_{1}+\left({\frac {\pi \vartheta _{3}(0,\tau )\vartheta _{4}(0,\tau )\vartheta _{2}(\pi z/\omega _{1},\tau )}{\omega _{1}\vartheta _{1}(\pi z/\omega _{1},\tau )}}\right)^{2}}" loading="lazy"></dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aae508e0d16371bbee7c6014aa84b19c71060381.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.637ex; height:5.009ex;" alt="{\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}" loading="lazy"> das Verhältnis der Perioden. Der Halbwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}=\wp ({\frac {\omega _{1}}{2}})}">

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<annotation encoding="application/x-tex">{\displaystyle e_{1}=\wp ({\frac {\omega _{1}}{2}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52e2a9894cd464411205dcfd06710de97e57a48b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.86ex; height:4.676ex;" alt="{\displaystyle e_{1}=\wp ({\frac {\omega _{1}}{2}})}" loading="lazy"> kann berechnet werden durch

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<annotation encoding="application/x-tex">{\displaystyle e_{1}={\frac {\pi ^{2}}{3\omega _{1}^{2}}}\left((\vartheta _{2}(0,\tau ))^{4}+2(\vartheta _{4}(0,\tau ))^{4}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7738878349875fc4e6aed0d1c991094715699b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.253ex; height:6.676ex;" alt="{\displaystyle e_{1}={\frac {\pi ^{2}}{3\omega _{1}^{2}}}\left((\vartheta _{2}(0,\tau ))^{4}+2(\vartheta _{4}(0,\tau ))^{4}\right)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Zusammenhang_mit_elliptischen_Kurven">Zusammenhang mit elliptischen Kurven</h2></div>

<div class="sieheauch" role="navigation" style="font-style:italic;">Siehe auch: <a href="Elliptische_Kurve" title="Elliptische Kurve">Elliptische Kurve</a></div>

Sei ein Gitter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3017b5b8e08ec4ff81bf287eb5c2cdbaba267be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.756ex; height:2.843ex;" alt="{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1},\omega _{2},g_{2},g_{3}\in \mathbb {C} \setminus \{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{1},\omega _{2},g_{2},g_{3}\in \mathbb {C} \setminus \{0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac9009d7ba56b62e776c5049cb5f963adf758521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.629ex; height:2.843ex;" alt="{\displaystyle \omega _{1},\omega _{2},g_{2},g_{3}\in \mathbb {C} \setminus \{0\}}" loading="lazy"> komplexe Zahlen sind, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b914a8bfef5d1b9b106048afa0aab4a99251f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}" loading="lazy"> linear unabhängig über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> sind.

Betrachte nun die ebene kubische Kurve

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<annotation encoding="application/x-tex">{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4430cfee774eeb483752672c90dd0fc2043d637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.042ex; height:3.343ex;" alt="{\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}}" loading="lazy"></dd></dl>

bzw. die <a href="Projektive_Geometrie" title="Projektive Geometrie">projektive</a> Kurve

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<annotation encoding="application/x-tex">{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}\cup \{\infty \}\subset \mathbb {P} _{\mathbb {C} }^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dfbed1b3f4f38d93c9764938ae1b56a03551000b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:57.315ex; height:4.176ex;" alt="{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x+g_{3}\}\cup \{\infty \}\subset \mathbb {P} _{\mathbb {C} }^{2}}" loading="lazy">.</dd></dl>

Für diese Kubiken, auch Weierstraßkubiken genannt, existieren keine Parametrisierungen durch rationale Funktionen, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta \neq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ecf280c732124f89f0a9a1095add4807f95e29b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.197ex; height:2.676ex;" alt="{\displaystyle \Delta \neq 0}" loading="lazy"> ist.<a href="#cite_note-:5-2">[2]</a> Trotzdem gibt es eine explizite Parametrisierung mittels der ℘-Funktion und ihrer Ableitung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp '}">

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<annotation encoding="application/x-tex">{\displaystyle \wp '}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98cc7b9ef41a255d70915227b703405b168cd8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.163ex; height:3.009ex;" alt="{\displaystyle \wp '}" loading="lazy">.

Damit erhält man die Abbildung

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<annotation encoding="application/x-tex">{\displaystyle \varphi :\mathbb {C} \setminus \Lambda \to C_{g_{2},g_{3}}^{\mathbb {C} },z\mapsto (\wp (z),\wp '(z))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21a84ee4148689c16ddf0a75ca721fda95a8f0de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.156ex; height:3.176ex;" alt="{\displaystyle \varphi :\mathbb {C} \setminus \Lambda \to C_{g_{2},g_{3}}^{\mathbb {C} },z\mapsto (\wp (z),\wp '(z))}" loading="lazy">.</dd></dl>

Indem man das Gitter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> auf den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

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<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy"> abbildet, kann die Abbildung fortgesetzt werden zu

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<annotation encoding="application/x-tex">{\displaystyle \varphi :\mathbb {C} \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9d8f989cf1e73d33469e79384b96358229604e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.437ex; height:4.176ex;" alt="{\displaystyle \varphi :\mathbb {C} \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}" loading="lazy">.</dd></dl>

Aufgrund der Periodizität von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp }">

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<annotation encoding="application/x-tex">{\displaystyle \wp }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4050ebf63686af152bf1ef5caabcdf2a2d812cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.478ex; height:2.176ex;" alt="{\displaystyle \wp }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp '}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \wp '}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98cc7b9ef41a255d70915227b703405b168cd8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.163ex; height:3.009ex;" alt="{\displaystyle \wp '}" loading="lazy"> ist diese Abbildung jedoch nicht injektiv. Wählt man stattdessen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} /\Lambda }">

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} /\Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9326cf3ac2c0ad37d1d33f02c3bbb927e0af65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.454ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} /\Lambda }" loading="lazy">, erhält man dann die Abbildung

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\varphi }}:\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b060e47f35d30e50d58fadc10b37c3db2b791243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.227ex; height:4.176ex;" alt="{\displaystyle {\tilde {\varphi }}:\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}" loading="lazy">.<a href="#cite_note-14">[14]</a></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} /\Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9326cf3ac2c0ad37d1d33f02c3bbb927e0af65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.454ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} /\Lambda }" loading="lazy"> ist dabei sowohl eine <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppe</a> als auch ein <a href="Topologischer_Raum" title="Topologischer Raum">topologischer Raum</a>, versehen mit der <a href="Quotiententopologie" title="Quotiententopologie">Quotiententopologie</a>.

Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\varphi }}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\varphi }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5e2e2e0ec24a538d31ae4da928b668ee8590469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.535ex; height:2.676ex;" alt="{\displaystyle {\tilde {\varphi }}}" loading="lazy"> ist nun bijektiv und parametrisiert die Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfdb73730f889bdbd41b104bfddba7fa054c7e6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.688ex; height:4.176ex;" alt="{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}" loading="lazy">.

Weiter lässt sich zeigen, dass jede glatte Weierstraßkubik auf diese Weise gegeben ist. Also dass es für jedes Paar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2},g_{3}\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle g_{2},g_{3}\in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e2a055c7778e8dc686739e4f1a416eb153a2fac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.879ex; height:2.509ex;" alt="{\displaystyle g_{2},g_{3}\in \mathbb {C} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}\neq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b9da3172b4de51c53e82b67b4cba51650a9ed87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.805ex; height:3.176ex;" alt="{\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}\neq 0}" loading="lazy"> ein Gitter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3017b5b8e08ec4ff81bf287eb5c2cdbaba267be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.756ex; height:2.843ex;" alt="{\displaystyle \Lambda :=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}" loading="lazy"> gibt, sodass

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<annotation encoding="application/x-tex">{\displaystyle g_{2}=g_{2}(\omega _{1},\omega _{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31c356cbaa0755497055ad01f0612c4cd88a295d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.268ex; height:2.843ex;" alt="{\displaystyle g_{2}=g_{2}(\omega _{1},\omega _{2})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}=g_{3}(\omega _{1},\omega _{2})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle g_{3}=g_{3}(\omega _{1},\omega _{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af1facfd64f39430753a9c5a2d0f5c813ae7208e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.268ex; height:2.843ex;" alt="{\displaystyle g_{3}=g_{3}(\omega _{1},\omega _{2})}" loading="lazy">.<a href="#cite_note-15">[15]</a>

Die Aussage, dass alle elliptischen Kurven über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> durch Modulformen über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> parametrisiert werden können, ist als <a href="Modularit%C3%A4tssatz" title="Modularitätssatz">Modularitätssatz</a> bekannt. Dieser Satz ist von großer Bedeutung in der <a href="Zahlentheorie" title="Zahlentheorie">Zahlentheorie</a>. <a href="Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a> konnte mit einem Teilbeweis des Modularitätssatzes 1995 den <a href="Gro%C3%9Fer_Fermatscher_Satz" title="Großer Fermatscher Satz">Großen Fermatschen Satz</a> beweisen.

<div class="mw-heading mw-heading2"><h2 id="Additionstheoreme">Additionstheoreme</h2></div>

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z,w\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle z,w\in \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4a0c53cf2374ff6e4b040d064d14550c50cafb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.305ex; height:2.509ex;" alt="{\displaystyle z,w\in \mathbb {C} }" loading="lazy">, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z,w,z+w,z-w\notin \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle z,w,z+w,z-w\notin \Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88ba83bfef753429b8c4d78de6cb9001f1098629.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.493ex; height:2.676ex;" alt="{\displaystyle z,w,z+w,z-w\notin \Lambda }" loading="lazy">. Dann gilt:<a href="#cite_note-:3-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle \wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/473c4e25c512a15485f6a89fc3669cf3c000882b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.868ex; height:7.009ex;" alt="{\displaystyle \wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w)}" loading="lazy">.</dd></dl>

Darüber hinaus gibt es noch die Verdopplungsformel:<a href="#cite_note-:3-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle \wp (2z)={\frac {1}{4}}\left[{\frac {\wp ''(z)}{\wp '(z)}}\right]^{2}-2\wp (z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5413dbf6a61b06ba58c4f84eeb0597d09b800161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.872ex; height:7.009ex;" alt="{\displaystyle \wp (2z)={\frac {1}{4}}\left[{\frac {\wp ''(z)}{\wp '(z)}}\right]^{2}-2\wp (z)}" loading="lazy">.</dd></dl>

Diese Formeln haben auch eine geometrische Bedeutung, wenn man wie im vorherigen Abschnitt die elliptische Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfdb73730f889bdbd41b104bfddba7fa054c7e6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.688ex; height:4.176ex;" alt="{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}" loading="lazy"> zusammen mit der Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\varphi }}:\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\varphi }}:\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b060e47f35d30e50d58fadc10b37c3db2b791243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.227ex; height:4.176ex;" alt="{\displaystyle {\tilde {\varphi }}:\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}" loading="lazy"> betrachtet.

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} /\Lambda }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9326cf3ac2c0ad37d1d33f02c3bbb927e0af65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.454ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} /\Lambda }" loading="lazy"> ist als <a href="Faktorgruppe" title="Faktorgruppe">Faktorgruppe</a> selbst eine Gruppe. Diese Gruppenstruktur überträgt sich auch auf die Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfdb73730f889bdbd41b104bfddba7fa054c7e6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.688ex; height:4.176ex;" alt="{\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}" loading="lazy"> (siehe <a href="Elliptische_Kurve#Gruppenoperation" title="Elliptische Kurve">Gruppenoperationen auf elliptischen Kurven</a>) und kann dort geometrisch interpretiert werden.

Damit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\varphi }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\varphi }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5e2e2e0ec24a538d31ae4da928b668ee8590469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.535ex; height:2.676ex;" alt="{\displaystyle {\tilde {\varphi }}}" loading="lazy"> dann insbesondere ein Gruppenisomorphismus<a href="#cite_note-:4-17">[17]</a>. Nun lässt sich das Additionstheorem auch auf folgende Weise geometrisch formulieren:

Die Summe dreier paarweise verschiedener Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle a,b,c\in {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aa2418f891a428e03eb2aba37c8018a680e6484.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.83ex; height:4.176ex;" alt="{\displaystyle a,b,c\in {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}" loading="lazy">ist genau dann Null, wenn sie auf einer gemeinsamen Geraden in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} _{\mathbb {C} }^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} _{\mathbb {C} }^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9c091df1177f69d191d5d138192b88aeb682a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.839ex; height:3.176ex;" alt="{\displaystyle \mathbb {P} _{\mathbb {C} }^{2}}" loading="lazy"> liegen<a href="#cite_note-:4-17">[17]</a>.

Dies ist äquivalent dazu, dass gilt:

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det \left({\begin{array}{rrr}1&\wp (u+v)&-\wp '(u+v)\\1&\wp (v)&\wp '(v)\\1&\wp (u)&\wp '(u)\\\end{array}}\right)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d2447df467457010e39425059d46e6cb9d6f7bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:38.167ex; height:9.843ex;" alt="{\displaystyle \det \left({\begin{array}{rrr}1&\wp (u+v)&-\wp '(u+v)\\1&\wp (v)&\wp '(v)\\1&\wp (u)&\wp '(u)\\\end{array}}\right)=0}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (u)=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp (u)=a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33df7ae9b8d2c4fbeb5d264e1ca2907262772f35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.946ex; height:2.843ex;" alt="{\displaystyle \wp (u)=a}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp (v)=b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wp (v)=b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2419dce962baacdad652e17518e4fd36b5ec969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle \wp (v)=b}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\notin \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u,v\notin \Lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edfa9c66d866eb5d18cad52403f39895e22e1935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.945ex; height:2.676ex;" alt="{\displaystyle u,v\notin \Lambda }" loading="lazy"> gelte.<a href="#cite_note-18">[18]</a>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="%C3%84quianharmonisch" title="Äquianharmonisch">äquianharmonischer Fall</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-:0-1"><a href="#cite_ref-:0_1-0">↑</a> Apostol, Tom M.: <cite style="font-style:italic">Modular functions and Dirichlet series in number theory</cite>. Springer-Verlag, New York 1976, ISBN 0-387-90185-X, S. 9.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Apostol%2C+Tom+M.&rft.btitle=Modular+functions+and+Dirichlet+series+in+number+theory&rft.date=1976&rft.genre=book&rft.isbn=038790185X&rft.pages=9&rft.place=New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-:5-2">↑ <a href="#cite_ref-:5_2-0">a</a> <a href="#cite_ref-:5_2-1">b</a> Hulek, Klaus.: <cite style="font-style:italic">Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen</cite>. 2., überarb. u. erw. Aufl. 2012. Vieweg+Teubner Verlag, Wiesbaden 2012, ISBN 978-3-8348-2348-9, S. 8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Hulek%2C+Klaus.&rft.btitle=Elementare+Algebraische+Geometrie+%3A+Grundlegende+Begriffe+und+Techniken+mit+zahlreichen+Beispielen+und+Anwendungen&rft.date=2012&rft.edition=2.%2C+%C3%BCberarb.+u.+erw.+Aufl.+2012&rft.genre=book&rft.isbn=9783834823489&rft.pages=8&rft.place=Wiesbaden&rft.pub=Vieweg%2BTeubner+Verlag" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Eberhard Freitag, Rolf Busam: <cite style="font-style:italic">Funktionentheorie 1</cite>. 4., korr. und erw. Auflage. Springer, Berlin 2006, ISBN 978-3-540-32058-6, S. 259.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Eberhard+Freitag%2C+Rolf+Busam&rft.btitle=Funktionentheorie+1&rft.date=2006&rft.edition=4.%2C+korr.+und+erw.+Aufl&rft.genre=book&rft.isbn=9783540320586&rft.pages=259&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Jeremy Gray: <cite style="font-style:italic">Real and the complex : a history of analysis in the 19th century</cite>. Cham 2015, ISBN 978-3-319-23715-2, S. 71.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Jeremy+Gray&rft.btitle=Real+and+the+complex+%3A+a+history+of+analysis+in+the+19th+century&rft.date=2015&rft.genre=book&rft.isbn=9783319237152&rft.pages=71&rft.place=Cham" style="display:none">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Eberhard Freitag, Rolf Busam: <cite style="font-style:italic">Funktionentheorie 1</cite>. 4., korr. und erw. Auflage. Springer, Berlin 2006, ISBN 978-3-540-32058-6, S. 294.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Eberhard+Freitag%2C+Rolf+Busam&rft.btitle=Funktionentheorie+1&rft.date=2006&rft.edition=4.%2C+korr.+und+erw.+Aufl&rft.genre=book&rft.isbn=9783540320586&rft.pages=294&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Eberhard Freitag, Rolf Busam: <cite style="font-style:italic">Funktionentheorie 1</cite>. 4., korr. und erw. Auflage. Springer, Berlin 2006, ISBN 978-3-540-32058-6, S. 268.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Eberhard+Freitag%2C+Rolf+Busam&rft.btitle=Funktionentheorie+1&rft.date=2006&rft.edition=4.%2C+korr.+und+erw.+Aufl&rft.genre=book&rft.isbn=9783540320586&rft.pages=268&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-:1-7">↑ <a href="#cite_ref-:1_7-0">a</a> <a href="#cite_ref-:1_7-1">b</a> <a href="#cite_ref-:1_7-2">c</a> <a href="#cite_ref-:1_7-3">d</a> Apostol, Tom M.: <cite style="font-style:italic">Modular functions and Dirichlet series in number theory</cite>. Springer-Verlag, New York 1976, ISBN 0-387-90185-X, S. 11.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Apostol%2C+Tom+M.&rft.btitle=Modular+functions+and+Dirichlet+series+in+number+theory&rft.date=1976&rft.genre=book&rft.isbn=038790185X&rft.pages=11&rft.place=New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-:2-8">↑ <a href="#cite_ref-:2_8-0">a</a> <a href="#cite_ref-:2_8-1">b</a> Apostol, Tom M.: <cite style="font-style:italic">Modular functions and Dirichlet series in number theory</cite>. Springer-Verlag, New York 1976, ISBN 0-387-90185-X, S. 14.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Apostol%2C+Tom+M.&rft.btitle=Modular+functions+and+Dirichlet+series+in+number+theory&rft.date=1976&rft.genre=book&rft.isbn=038790185X&rft.pages=14&rft.place=New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Eberhard Freitag, Rolf Busam: <cite style="font-style:italic">Funktionentheorie 1</cite>. 4., korr. und erw. Auflage. Springer, Berlin 2006, ISBN 978-3-540-32058-6, S. 270.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Eberhard+Freitag%2C+Rolf+Busam&rft.btitle=Funktionentheorie+1&rft.date=2006&rft.edition=4.%2C+korr.+und+erw.+Aufl&rft.genre=book&rft.isbn=9783540320586&rft.pages=270&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Tom M. Apostol: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Modular functions and Dirichlet series in number theory</cite>. Springer, New York 1976, ISBN 0-387-90185-X, S. 13 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Tom+M.+Apostol&rft.btitle=Modular+functions+and+Dirichlet+series+in+number+theory&rft.date=1976&rft.genre=book&rft.isbn=038790185X&rft.pages=13&rft.place=New+York&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> K. Chandrasekharan: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Elliptic functions</cite>. Springer, Berlin 1985, ISBN 0-387-15295-4, S. 33 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=K.+Chandrasekharan&rft.btitle=Elliptic+functions&rft.date=1985&rft.genre=book&rft.isbn=0387152954&rft.pages=33&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> Theresa M. Korn: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Mathematical Handbook for Scientists and Engineers</cite>. McGraw–Hill, New York 1961, S. 721 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Theresa+M.+Korn&rft.btitle=Mathematical+Handbook+for+Scientists+and+Engineers&rft.date=1961&rft.genre=book&rft.pages=721&rft.place=New+York&rft.pub=McGraw-Hill" style="display:none">

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> <a rel="nofollow" class="external text" href="https://dlmf.nist.gov/23.6">NIST Digital Library of Mathematical Functions.</a> Abgerufen am 21. Oktober 2025 (Bezeichnungen angepasst).<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AWeierstra%C3%9Fsche+%E2%84%98-Funktion&rft.title=NIST+Digital+Library+of+Mathematical+Functions&rft.description=NIST+Digital+Library+of+Mathematical+Functions&rft.identifier=https%3A%2F%2Fdlmf.nist.gov%2F23.6">

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> Hulek, Klaus.: <cite style="font-style:italic">Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen</cite>. 2., überarb. u. erw. Aufl. 2012. Vieweg+Teubner Verlag, Wiesbaden 2012, ISBN 978-3-8348-2348-9, S. 12.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Hulek%2C+Klaus.&rft.btitle=Elementare+Algebraische+Geometrie+%3A+Grundlegende+Begriffe+und+Techniken+mit+zahlreichen+Beispielen+und+Anwendungen&rft.date=2012&rft.edition=2.%2C+%C3%BCberarb.+u.+erw.+Aufl.+2012&rft.genre=book&rft.isbn=9783834823489&rft.pages=12&rft.place=Wiesbaden&rft.pub=Vieweg%2BTeubner+Verlag" style="display:none">

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> Hulek, Klaus.: <cite style="font-style:italic">Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen</cite>. 2., überarb. u. erw. Aufl. 2012. Vieweg+Teubner Verlag, Wiesbaden 2012, ISBN 978-3-8348-2348-9, S. 111.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Hulek%2C+Klaus.&rft.btitle=Elementare+Algebraische+Geometrie+%3A+Grundlegende+Begriffe+und+Techniken+mit+zahlreichen+Beispielen+und+Anwendungen&rft.date=2012&rft.edition=2.%2C+%C3%BCberarb.+u.+erw.+Aufl.+2012&rft.genre=book&rft.isbn=9783834823489&rft.pages=111&rft.place=Wiesbaden&rft.pub=Vieweg%2BTeubner+Verlag" style="display:none">

</li>

<li id="cite_note-:3-16">↑ <a href="#cite_ref-:3_16-0">a</a> <a href="#cite_ref-:3_16-1">b</a> Eberhard Freitag, Rolf Busam: <cite style="font-style:italic">Funktionentheorie 1</cite>. 4., korr. und erw. Auflage. Springer, Berlin 2006, ISBN 978-3-540-32058-6, S. 286.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Eberhard+Freitag%2C+Rolf+Busam&rft.btitle=Funktionentheorie+1&rft.date=2006&rft.edition=4.%2C+korr.+und+erw.+Aufl&rft.genre=book&rft.isbn=9783540320586&rft.pages=286&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-:4-17">↑ <a href="#cite_ref-:4_17-0">a</a> <a href="#cite_ref-:4_17-1">b</a> Eberhard Freitag, Rolf Busam: <cite style="font-style:italic">Funktionentheorie 1</cite>. 4., korr. und erw. Auflage. Springer, Berlin 2006, ISBN 978-3-540-32058-6, S. 287.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Eberhard+Freitag%2C+Rolf+Busam&rft.btitle=Funktionentheorie+1&rft.date=2006&rft.edition=4.%2C+korr.+und+erw.+Aufl&rft.genre=book&rft.isbn=9783540320586&rft.pages=287&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-18"><a href="#cite_ref-18">↑</a> Eberhard Freitag, Rolf Busam: <cite style="font-style:italic">Funktionentheorie 1</cite>. 4., korr. und erw. Auflage. Springer, Berlin 2006, ISBN 978-3-540-32058-6, S. 288.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9Fsche+%E2%84%98-Funktion&rft.au=Eberhard+Freitag%2C+Rolf+Busam&rft.btitle=Funktionentheorie+1&rft.date=2006&rft.edition=4.%2C+korr.+und+erw.+Aufl&rft.genre=book&rft.isbn=9783540320586&rft.pages=288&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

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