ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Quaternionic analysis</title>

</head>

Quaternionic analysis

</h1>

</header>

In <a href="Mathematics" title="Mathematics">mathematics</a>, quaternionic analysis is the study of <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> with <a href="Quaternion" title="Quaternion">quaternions</a> as the <a href="Domain_of_a_function" title="Domain of a function">domain</a> and/or range. Such functions can be called functions of a quaternion variable just as <a href="Function_of_a_real_variable" title="Function of a real variable">functions of a real variable</a> or a <a href="Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable">complex variable</a> are called.

As with <a href="Complex_analysis" title="Complex analysis">complex</a> and <a href="Real_analysis" title="Real analysis">real analysis</a>, it is possible to study the concepts of <a href="Analytic_function" title="Analytic function">analyticity</a>, <a href="Holomorphic_function" title="Holomorphic function">holomorphy</a>, <a href="Harmonic_function" title="Harmonic function">harmonicity</a> and <a href="Conformality" class="mw-redirect" title="Conformality">conformality</a> in the context of quaternions. Unlike the <a href="Complex_number" title="Complex number">complex numbers</a> and like the <a href="Real_number" title="Real number">reals</a>, the four notions do not coincide.

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

The <a href="Projection_(linear_algebra)" title="Projection (linear algebra)">projections</a> of a quaternion onto its scalar part or onto its vector part, as well as the modulus and <a href="Versor" title="Versor">versor</a> functions, are examples that are basic to understanding quaternion structure.

An important example of a function of a quaternion variable is

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}(q)=uqu^{-1}}</annotation>

</semantics>

</math><img src="./5ab9c234103bb1d7aa3204bd23d84d6b35e93420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.232ex; height:3.176ex;" alt="{\displaystyle f_{1}(q)=uqu^{-1}}" loading="lazy"></dd></dl>

which <a href="Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">rotates the vector part of q</a> by twice the angle represented by the versor u.

The quaternion <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}(q)=q^{-1}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}(q)=q^{-1}}</annotation>

</semantics>

</math><img src="./bbf6b556722fbe0d327e40330206cec5b7713907.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.583ex; height:3.176ex;" alt="{\displaystyle f_{2}(q)=q^{-1}}" loading="lazy"> is another fundamental function, but as with other number systems, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}(0)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{2}(0)}</annotation>

</semantics>

</math><img src="./2652df6f0722780b3fa8ee9194a85d14181e1a04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.165ex; height:2.843ex;" alt="{\displaystyle f_{2}(0)}" loading="lazy"> and related problems are generally excluded due to the nature of <a href="Dividing_by_zero" class="mw-redirect" title="Dividing by zero">dividing by zero</a>.

<a href="Affine_transformation" title="Affine transformation">Affine transformations</a> of quaternions have the form

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{3}(q)=aq+b,\quad a,b,q\in \mathbb {H} .}</annotation>

</semantics>

</math><img src="./be1be6ece1fd603578c123d348fe4c9d09849700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.325ex; height:2.843ex;" alt="{\displaystyle f_{3}(q)=aq+b,\quad a,b,q\in \mathbb {H} .}" loading="lazy"></dd></dl>

<a href="Linear_fractional_transformation" title="Linear fractional transformation">Linear fractional transformations</a> of quaternions can be represented by elements of the <a href="Matrix_ring" title="Matrix ring">matrix ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}(\mathbb {H} )}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{2}(\mathbb {H} )}</annotation>

</semantics>

</math><img src="./d5f1b47e0a9080f09cc5f530fa3eca76bf7730d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.926ex; height:2.843ex;" alt="{\displaystyle M_{2}(\mathbb {H} )}" loading="lazy"> operating on the <a href="Projective_line_over_a_ring" title="Projective line over a ring">projective line over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e050965453c42bcc6bd544546703c836bdafeac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" loading="lazy"></a>. For instance, the mappings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\mapsto uqv,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\mapsto uqv,}</annotation>

</semantics>

</math><img src="./38ba91d54a480c8c7f55cbf1207acd533d44c904.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.857ex; height:2.176ex;" alt="{\displaystyle q\mapsto uqv,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> are fixed <a href="Versor" title="Versor">versors</a> serve to produce the <a href="Elliptic_geometry#elliptic_space" title="Elliptic geometry">motions of elliptic space</a>.

Quaternion variable theory differs in some respects from complex variable theory. For example: The <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, <a href="Holomorphic_function" title="Holomorphic function">non-analytic</a> operation. Indeed, conjugation changes the <a href="Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a> of plane figures, something that arithmetic functions do not change.

In contrast to the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a>, the quaternion conjugation can be expressed arithmetically, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{4}(q)=-{\tfrac {1}{2}}(q+iqi+jqj+kqk)}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{4}(q)=-{\tfrac {1}{2}}(q+iqi+jqj+kqk)}</annotation>

</semantics>

</math><img src="./4ac5f4b271448fe04e8f84881564a983ab2e6795.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.189ex; height:3.509ex;" alt="{\displaystyle f_{4}(q)=-{\tfrac {1}{2}}(q+iqi+jqj+kqk)}" loading="lazy">

This equation can be proven, starting with the <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> {1, i, j, k}:

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{4}(1)=-{\tfrac {1}{2}}(1-1-1-1)=1,\quad f_{4}(i)=-{\tfrac {1}{2}}(i-i+i+i)=-i,\quad f_{4}(j)=-j,\quad f_{4}(k)=-k}</annotation>

</semantics>

</math><img src="./592465cd67c54492071c4f013d5d95300f9b6870.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:93.817ex; height:3.509ex;" alt="{\displaystyle f_{4}(1)=-{\tfrac {1}{2}}(1-1-1-1)=1,\quad f_{4}(i)=-{\tfrac {1}{2}}(i-i+i+i)=-i,\quad f_{4}(j)=-j,\quad f_{4}(k)=-k}" loading="lazy">.</dd></dl>

Consequently, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{4}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{4}}</annotation>

</semantics>

</math><img src="./294f14bcf91a6ff040c4c22b71720764d800fc26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{4}}" loading="lazy"> is <a href="Linear_map" title="Linear map">linear</a>,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{4}(q)=f_{4}(w+xi+yj+zk)=wf_{4}(1)+xf_{4}(i)+yf_{4}(j)+zf_{4}(k)=w-xi-yj-zk=q^{*}.}</annotation>

</semantics>

</math><img src="./054090c50ce112ebbf5f6655758f3b9359f39ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:91.614ex; height:2.843ex;" alt="{\displaystyle f_{4}(q)=f_{4}(w+xi+yj+zk)=wf_{4}(1)+xf_{4}(i)+yf_{4}(j)+zf_{4}(k)=w-xi-yj-zk=q^{*}.}" loading="lazy"></dd></dl>

The success of <a href="Complex_analysis" title="Complex analysis">complex analysis</a> in providing a rich family of <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a> for scientific work has engaged some workers in efforts to extend the planar theory, based on complex numbers, to a 4-space study with functions of a quaternion variable.<a href="#cite_note-1">[1]</a> These efforts were summarized in <a href="#CITEREFDeavours1973">Deavours (1973)</a>.<a href="#cite_note-2">[a]</a>

Though <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e050965453c42bcc6bd544546703c836bdafeac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" loading="lazy"> <a href="Quaternion#As_a_union_of_complex_planes" title="Quaternion">appears as a union of complex planes</a>, the following proposition shows that extending complex functions requires special care:

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{5}(z)=u(x,y)+iv(x,y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{5}(z)=u(x,y)+iv(x,y)}</annotation>

</semantics>

</math><img src="./d7ee0e94c11fa21f1a68ef347a4d17fbcb8ff8ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.946ex; height:2.843ex;" alt="{\displaystyle f_{5}(z)=u(x,y)+iv(x,y)}" loading="lazy"> be a function of a complex variable, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./08e90bb6b36fef59c6113eed2a08f10d77240741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}" loading="lazy">. Suppose also that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is an <a href="Even_function" class="mw-redirect" title="Even function">even function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is an <a href="Odd_function" class="mw-redirect" title="Odd function">odd function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{5}(q)=u(x,y)+rv(x,y)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{5}(q)=u(x,y)+rv(x,y)}</annotation>

</semantics>

</math><img src="./3e6678e31f3983f9a6724dd332fe326aea4544ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.174ex; height:2.843ex;" alt="{\displaystyle f_{5}(q)=u(x,y)+rv(x,y)}" loading="lazy"> is an extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{5}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{5}}</annotation>

</semantics>

</math><img src="./77f823adf7075ef3765c1feae778fa971be865a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{5}}" loading="lazy"> to a quaternion variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=x+yr}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=x+yr}</annotation>

</semantics>

</math><img src="./725168613c34368078cdd7e23ff61b2002783687.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.542ex; height:2.343ex;" alt="{\displaystyle q=x+yr}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}=-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5c56ed2bf20aa46005cb12a4f4859152cfd7fc61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.172ex; height:2.843ex;" alt="{\displaystyle r^{2}=-1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in \mathbb {H} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./84ab8ef83b17f67c73d23e5a17d7fe073d4b680b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.697ex; height:2.176ex;" alt="{\displaystyle r\in \mathbb {H} }" loading="lazy">.

Then, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cc488e611bcc916d2da5dec54181e4909297e088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.103ex; height:2.343ex;" alt="{\displaystyle r^{*}}" loading="lazy"> represent the conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=x-yr^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=x-yr^{*}}</annotation>

</semantics>

</math><img src="./bf6230d313529adb814328f3c1e498183106570c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.596ex; height:2.676ex;" alt="{\displaystyle q=x-yr^{*}}" loading="lazy">. The extension to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e050965453c42bcc6bd544546703c836bdafeac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" loading="lazy"> will be complete when it is shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{5}(q)=f_{5}(x-yr^{*})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{5}(q)=f_{5}(x-yr^{*})}</annotation>

</semantics>

</math><img src="./fa70d4ad6444451e73b0bcb15d118e1720eda90c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.602ex; height:2.843ex;" alt="{\displaystyle f_{5}(q)=f_{5}(x-yr^{*})}" loading="lazy">. Indeed, by hypothesis

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(x,y)=u(x,-y),\quad v(x,y)=-v(x,-y)\quad }</annotation>

</semantics>

</math><img src="./5e8f316b37adf4c7cbce3c18827a3d5680445aef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.529ex; height:2.843ex;" alt="{\displaystyle u(x,y)=u(x,-y),\quad v(x,y)=-v(x,-y)\quad }" loading="lazy"> one obtains</dd>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{5}(x-yr^{*})=u(x,-y)+r^{*}v(x,-y)=u(x,y)+rv(x,y)=f_{5}(q).}</annotation>

</semantics>

</math><img src="./339933484e3167fb4b25568f325ad993e5f98c7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.123ex; height:2.843ex;" alt="{\displaystyle f_{5}(x-yr^{*})=u(x,-y)+r^{*}v(x,-y)=u(x,y)+rv(x,y)=f_{5}(q).}" loading="lazy"></dd></dl>

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

In the following, colons and square brackets are used to denote <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous vectors</a>.

The <a href="Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">rotation</a> about axis r is a classical application of quaternions to <a href="Space" title="Space">space</a> mapping.<a href="#cite_note-3">[2]</a>

In terms of a <a href="Homography#Over_a_ring" title="Homography">homography</a>, the rotation is expressed

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [q:1]{\begin{pmatrix}u&0\\0&u\end{pmatrix}}=[qu:u]\thicksim [u^{-1}qu:1],}</annotation>

</semantics>

</math><img src="./073587481c4dc543f6a8f46326542530946fd4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.876ex; height:6.176ex;" alt="{\displaystyle [q:1]{\begin{pmatrix}u&0\\0&u\end{pmatrix}}=[qu:u]\thicksim [u^{-1}qu:1],}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=\exp(\theta r)=\cos \theta +r\sin \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=\exp(\theta r)=\cos \theta +r\sin \theta }</annotation>

</semantics>

</math><img src="./0ef8e8e5b8bc9e9b677f9c58cf0737ef7dd177a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.225ex; height:2.843ex;" alt="{\displaystyle u=\exp(\theta r)=\cos \theta +r\sin \theta }" loading="lazy"> is a <a href="Versor" title="Versor">versor</a>. If p * = −p, then the translation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\mapsto q+p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\mapsto q+p}</annotation>

</semantics>

</math><img src="./44152f51f487584a2157c87c97a65c5cd743c6cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.763ex; height:2.343ex;" alt="{\displaystyle q\mapsto q+p}" loading="lazy"> is expressed by

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [q:1]{\begin{pmatrix}1&0\\p&1\end{pmatrix}}=[q+p:1].}</annotation>

</semantics>

</math><img src="./84a7c8113e9e19a2489f5e96f4c1f7ff24f7e0fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.508ex; height:6.176ex;" alt="{\displaystyle [q:1]{\begin{pmatrix}1&0\\p&1\end{pmatrix}}=[q+p:1].}" loading="lazy"></dd></dl>

Rotation and translation xr along the axis of rotation is given by

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [q:1]{\begin{pmatrix}u&0\\uxr&u\end{pmatrix}}=[qu+uxr:u]\thicksim [u^{-1}qu+xr:1].}</annotation>

</semantics>

</math><img src="./5517a98807ade53093dae41200aa3796a6c8c3af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.021ex; height:6.176ex;" alt="{\displaystyle [q:1]{\begin{pmatrix}u&0\\uxr&u\end{pmatrix}}=[qu+uxr:u]\thicksim [u^{-1}qu+xr:1].}" loading="lazy"></dd></dl>

Such a mapping is called a <a href="Screw_displacement" class="mw-redirect" title="Screw displacement">screw displacement</a>. In classical <a href="Kinematics" title="Kinematics">kinematics</a>, <a href="Chasles'_theorem_(kinematics)" title="Chasles' theorem (kinematics)">Chasles' theorem</a> states that any rigid body motion can be displayed as a screw displacement. Just as the representation of a <a href="Euclidean_plane_isometry" title="Euclidean plane isometry">Euclidean plane isometry</a> as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and the <a href="Screw_axis" title="Screw axis">screw axis</a> required, is a matter of quaternion arithmetic with homographies: Let s be a right versor, or square root of minus one, perpendicular to r, with t = rs.

Consider the axis passing through s and parallel to r. Rotation about it is expressed<a href="#cite_note-4">[3]</a> by the homography composition

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&0\\-s&1\end{pmatrix}}{\begin{pmatrix}u&0\\0&u\end{pmatrix}}{\begin{pmatrix}1&0\\s&1\end{pmatrix}}={\begin{pmatrix}u&0\\z&u\end{pmatrix}},}</annotation>

</semantics>

</math><img src="./81719583b652578d90cd697cafc2dc79272fda21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.431ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&0\\-s&1\end{pmatrix}}{\begin{pmatrix}u&0\\0&u\end{pmatrix}}{\begin{pmatrix}1&0\\s&1\end{pmatrix}}={\begin{pmatrix}u&0\\z&u\end{pmatrix}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=us-su=\sin \theta (rs-sr)=2t\sin \theta .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=us-su=\sin \theta (rs-sr)=2t\sin \theta .}</annotation>

</semantics>

</math><img src="./eac5c189832af401a844d0502ac9260ffc492b2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.694ex; height:2.843ex;" alt="{\displaystyle z=us-su=\sin \theta (rs-sr)=2t\sin \theta .}" loading="lazy">

Now in the (s,t)-plane the parameter θ traces out a circle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{-1}z=u^{-1}(2t\sin \theta )=2\sin \theta (t\cos \theta -s\sin \theta )}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u^{-1}z=u^{-1}(2t\sin \theta )=2\sin \theta (t\cos \theta -s\sin \theta )}</annotation>

</semantics>

</math><img src="./8c37c943b3df01e4b0eadebf98e9158236cdf877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.3ex; height:3.176ex;" alt="{\displaystyle u^{-1}z=u^{-1}(2t\sin \theta )=2\sin \theta (t\cos \theta -s\sin \theta )}" loading="lazy"> in the half-plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace wt+xs:x>0\rbrace .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lbrace wt+xs:x>0\rbrace .}</annotation>

</semantics>

</math><img src="./f2193afa3256ae8941f53773f905103c232563ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.264ex; height:2.843ex;" alt="{\displaystyle \lbrace wt+xs:x>0\rbrace .}" loading="lazy">

Any p in this half-plane lies on a ray from the origin through the circle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lbrace u^{-1}z:0<\theta <\pi \rbrace }">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lbrace u^{-1}z:0<\theta <\pi \rbrace }</annotation>

</semantics>

</math><img src="./f9705306a67702e53765e21e30289826441fe58b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.795ex; height:3.176ex;" alt="{\displaystyle \lbrace u^{-1}z:0<\theta <\pi \rbrace }" loading="lazy"> and can be written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=au^{-1}z,\ \ a>0.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=au^{-1}z,\ \ a>0.}</annotation>

</semantics>

</math><img src="./4d20f98b36a6cb9354e5b409a6823724c940797a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:18.671ex; height:3.009ex;" alt="{\displaystyle p=au^{-1}z,\ \ a>0.}" loading="lazy">

Then up = az, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}u&0\\az&u\end{pmatrix}}}">

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}u&0\\az&u\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./60493484bd99dc5fe6f69b0e1b6e167886d6e34e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.143ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}u&0\\az&u\end{pmatrix}}}" loading="lazy"> as the homography expressing <a href="Conjugation_(group_theory)" class="mw-redirect" title="Conjugation (group theory)">conjugation</a> of a rotation by a translation p.

<div class="mw-heading mw-heading2"><h2 id="The_derivative_for_quaternions">The derivative for quaternions</h2></div>

Since the time of Hamilton, it has been realized that requiring the independence of the <a href="Derivative" title="Derivative">derivative</a> from the path that a differential follows toward zero is too restrictive: it excludes even <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(q)=q^{2}\ }">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ f(q)=q^{2}\ }</annotation>

</semantics>

</math><img src="./84c36e8369a07f8629cba36bcb08f3c5bfdf2b47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.551ex; height:3.176ex;" alt="{\displaystyle \ f(q)=q^{2}\ }" loading="lazy"> from differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.<a href="#cite_note-5">[4]</a><a href="#cite_note-6">[5]</a>

Considering the increment of <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> of quaternionic argument shows that the increment is a linear map of increment of the argument. From this, a definition can be made:

A continuous function

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ f:\mathbb {H} \rightarrow \mathbb {H} \ }</annotation>

</semantics>

</math><img src="./b4791a408da02e38616442c7f3f7f57816a72805.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.607ex; height:2.509ex;" alt="{\displaystyle \ f:\mathbb {H} \rightarrow \mathbb {H} \ }" loading="lazy">

is called differentiable on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ U\subset \mathbb {H} \ ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ U\subset \mathbb {H} \ ,}</annotation>

</semantics>

</math><img src="./29345f44af01e1679a3ab87551c137e5d2a94ce4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.497ex; height:2.509ex;" alt="{\displaystyle \ U\subset \mathbb {H} \ ,}" loading="lazy"> if at every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x\in U\ ,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cc82a1715946a60bbf83670736f888952f960b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.761ex; height:2.509ex;" alt="{\displaystyle \ x\in U\ ,}" loading="lazy"> an increment of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f\ }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4a7cd8f2c9f7c532472574d7a1713be631b0f77a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.44ex; height:2.509ex;" alt="{\displaystyle \ f\ }" loading="lazy"> corresponding to a quaternion increment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ h\ }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5e28abe4dc17e614fad3a5a46c0fa71186dc7f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.5ex; height:2.176ex;" alt="{\displaystyle \ h\ }" loading="lazy"> of its argument, can be represented as

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+h)-f(x)={\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ h+o(h)}</annotation>

</semantics>

</math><img src="./4505c54c3feb3385577edbde6413b1b7f80a7961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.537ex; height:5.843ex;" alt="{\displaystyle f(x+h)-f(x)={\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ h+o(h)}" loading="lazy"></dd></dl>

where

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}:\mathbb {H} \rightarrow \mathbb {H} }</annotation>

</semantics>

</math><img src="./8b609fc320ea95604dde8f3e8d4b93c895780eee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.101ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}:\mathbb {H} \rightarrow \mathbb {H} }" loading="lazy"></dd></dl>

is <a href="Linear_map" title="Linear map">linear map</a> of quaternion algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mathbb {H} \ ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4e96822c56186a1059d44fda58b9952764ac9570.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle \ \mathbb {H} \ ,}" loading="lazy"> and

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ o:\mathbb {H} \rightarrow \mathbb {H} \ }</annotation>

</semantics>

</math><img src="./0986ef80f3d26ee3c79f1b1305a47822835c74ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.456ex; height:2.176ex;" alt="{\displaystyle \ o:\mathbb {H} \rightarrow \mathbb {H} \ }" loading="lazy">

represents some continuous map such that

</mrow>

</munder>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{a\rightarrow 0}{\frac {\ \left|\ o(a)\ \right|\ }{\left|\ a\ \right|}}=0\ ,}</annotation>

</semantics>

</math><img src="./110bfb8dd8a4a396dad442051dde811c7e1717c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.604ex; height:6.509ex;" alt="{\displaystyle \lim _{a\rightarrow 0}{\frac {\ \left|\ o(a)\ \right|\ }{\left|\ a\ \right|}}=0\ ,}" loading="lazy"></dd></dl>

and the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \circ h\ }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./85f783db410b0c5f3afa6a0efda8c6232ab0d99e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.695ex; height:2.176ex;" alt="{\displaystyle \ \circ h\ }" loading="lazy"> denotes ...

The linear map

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}}</annotation>

</semantics>

</math><img src="./4c58119fdc5f173af1c1bcdfcd5c1e978641b35f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.933ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}}" loading="lazy">

is called the derivative of the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f~.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ f~.}</annotation>

</semantics>

</math><img src="./b1031a7321e6383529d984f8461a2348076137d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.087ex; height:2.509ex;" alt="{\displaystyle \ f~.}" loading="lazy">

On the quaternions, the derivative may be expressed as

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}=\sum _{s}{\frac {\operatorname {d} _{s0}f(x)}{\operatorname {d} x}}\otimes {\frac {\operatorname {d} _{s1}f(x)}{\operatorname {d} x}}}</annotation>

</semantics>

</math><img src="./ea599e5fa7862897cf394a4e0fb22cc5f37162c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.131ex; height:6.843ex;" alt="{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}=\sum _{s}{\frac {\operatorname {d} _{s0}f(x)}{\operatorname {d} x}}\otimes {\frac {\operatorname {d} _{s1}f(x)}{\operatorname {d} x}}}" loading="lazy"></dd></dl>

Therefore, the differential of the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f\ }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4a7cd8f2c9f7c532472574d7a1713be631b0f77a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.44ex; height:2.509ex;" alt="{\displaystyle \ f\ }" loading="lazy"> may be expressed as follows, with brackets on either side.

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ \operatorname {d} x=\left(\sum _{s}{\frac {\operatorname {d} _{s0}f(x)}{\operatorname {d} x}}\otimes {\frac {\operatorname {d} _{s1}f(x)}{\operatorname {d} x}}\right)\circ \operatorname {d} x=\sum _{s}{\frac {\operatorname {d} _{s0}f(x)}{\operatorname {d} x}}\left(\operatorname {d} x\right){\frac {\operatorname {d} _{s1}f(x)}{\operatorname {d} x}}}</annotation>

</semantics>

</math><img src="./ff76ad5f2951821b9546a42e71929a5a0a95f8fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.171ex; height:7.509ex;" alt="{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ \operatorname {d} x=\left(\sum _{s}{\frac {\operatorname {d} _{s0}f(x)}{\operatorname {d} x}}\otimes {\frac {\operatorname {d} _{s1}f(x)}{\operatorname {d} x}}\right)\circ \operatorname {d} x=\sum _{s}{\frac {\operatorname {d} _{s0}f(x)}{\operatorname {d} x}}\left(\operatorname {d} x\right){\frac {\operatorname {d} _{s1}f(x)}{\operatorname {d} x}}}" loading="lazy"></dd></dl>

The number of terms in the sum will depend on the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f~.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ f~.}</annotation>

</semantics>

</math><img src="./b1031a7321e6383529d984f8461a2348076137d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.087ex; height:2.509ex;" alt="{\displaystyle \ f~.}" loading="lazy"> The expressions

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ~~{\frac {\operatorname {d} _{sp}\operatorname {d} f(x)}{\operatorname {d} x}}~~{\mathsf {\ for\ }}~~p=0,1~~}</annotation>

</semantics>

</math><img src="./1861db9207c91b1e8e821c60b2663d6834cbf0d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.641ex; height:5.843ex;" alt="{\displaystyle ~~{\frac {\operatorname {d} _{sp}\operatorname {d} f(x)}{\operatorname {d} x}}~~{\mathsf {\ for\ }}~~p=0,1~~}" loading="lazy"> are called

components of derivative.

The derivative of a quaternionic function is defined by the expression

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ h=\lim _{t\to 0}\left(\ {\frac {\ f(x+t\ h)-f(x)\ }{t}}\ \right)}</annotation>

</semantics>

</math><img src="./cc0ea83054580bfb838a937a178072e74c790768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.037ex; height:6.343ex;" alt="{\displaystyle {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ h=\lim _{t\to 0}\left(\ {\frac {\ f(x+t\ h)-f(x)\ }{t}}\ \right)}" loading="lazy"></dd></dl>

where the variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ t\ }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b0e0bb2a6a3d4150f3bfad42d365078faa5e97be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.001ex; height:2.009ex;" alt="{\displaystyle \ t\ }" loading="lazy"> is a real scalar.

The following equations then hold:

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} \left(f(x)+g(x)\right)}{\operatorname {d} x}}={\frac {\operatorname {d} f(x)}{\operatorname {d} x}}+{\frac {\operatorname {d} g(x)}{\operatorname {d} x}}}</annotation>

</semantics>

</math><img src="./15d17b8e170a5508eacaeb0a0def35c998018ba3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.094ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} \left(f(x)+g(x)\right)}{\operatorname {d} x}}={\frac {\operatorname {d} f(x)}{\operatorname {d} x}}+{\frac {\operatorname {d} g(x)}{\operatorname {d} x}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} \left(f(x)\ g(x)\right)}{\operatorname {d} x}}={\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\ g(x)+f(x)\ {\frac {\operatorname {d} g(x)}{\operatorname {d} x}}}</annotation>

</semantics>

</math><img src="./e84d40d3b89eee93695905fec1eede794a08bc28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.668ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} \left(f(x)\ g(x)\right)}{\operatorname {d} x}}={\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\ g(x)+f(x)\ {\frac {\operatorname {d} g(x)}{\operatorname {d} x}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} \left(f(x)\ g(x)\right)}{\operatorname {d} x}}\circ h=\left({\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ h\right)\ g(x)+f(x)\left({\frac {\operatorname {d} g(x)}{\operatorname {d} x}}\circ h\right)}</annotation>

</semantics>

</math><img src="./d04ebce67e1daaf0db7c717df0a7ee7aa633692d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.305ex; height:6.343ex;" alt="{\displaystyle {\frac {\operatorname {d} \left(f(x)\ g(x)\right)}{\operatorname {d} x}}\circ h=\left({\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ h\right)\ g(x)+f(x)\left({\frac {\operatorname {d} g(x)}{\operatorname {d} x}}\circ h\right)}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} \left(a\ f(x)\ b\right)}{\operatorname {d} x}}=a\ {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\ b}</annotation>

</semantics>

</math><img src="./5614b41782110fe54f1617f18d9be46c6fec71c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.165ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} \left(a\ f(x)\ b\right)}{\operatorname {d} x}}=a\ {\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\ b}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} \left(a\ f(x)\ b\right)}{\operatorname {d} x}}\circ h=a\left({\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ h\right)b}</annotation>

</semantics>

</math><img src="./a35cc37902e9151aa9e82ed266172fa5175599e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.266ex; height:6.343ex;" alt="{\displaystyle {\frac {\operatorname {d} \left(a\ f(x)\ b\right)}{\operatorname {d} x}}\circ h=a\left({\frac {\operatorname {d} f(x)}{\operatorname {d} x}}\circ h\right)b}" loading="lazy"></dd></dl>

For the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=a\ x\ b\ ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ f(x)=a\ x\ b\ ,}</annotation>

</semantics>

</math><img src="./7e0f6bd36208fa227f7b5c7fcac6f5edef3f02af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.043ex; height:2.843ex;" alt="{\displaystyle \ f(x)=a\ x\ b\ ,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ a\ }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8124de742ae987fe73be9ca9d3d4ba8586e28b11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.391ex; height:1.676ex;" alt="{\displaystyle \ a\ }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ b\ }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./978008481f65e59642b3c5cf1ad23b84284fff31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.159ex; height:2.176ex;" alt="{\displaystyle \ b\ }" loading="lazy"> are constant quaternions, the derivative is

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} \left(a\ x\ b\right)}{\operatorname {d} x}}=a\otimes b}</annotation>

</semantics>

</math><img src="./5621b7e25eb3127d84fb0385f2d3066e19ca8f59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.823ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} \left(a\ x\ b\right)}{\operatorname {d} x}}=a\otimes b}" loading="lazy">

</td>

</td>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {d} y={\frac {\operatorname {d} \left(a\ x\ b\right)}{\operatorname {d} x}}\circ \operatorname {d} x=a\ \left(\operatorname {d} x\right)\ b}</annotation>

</semantics>

</math><img src="./468accc67848b203824553df1a90b2313a976607.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.874ex; height:5.843ex;" alt="{\displaystyle \operatorname {d} y={\frac {\operatorname {d} \left(a\ x\ b\right)}{\operatorname {d} x}}\circ \operatorname {d} x=a\ \left(\operatorname {d} x\right)\ b}" loading="lazy">

</td></tr></tbody></table>

and so the components are:

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} _{10}\left(a\ x\ b\right)}{\operatorname {d} x}}=a}</annotation>

</semantics>

</math><img src="./8add78f92403dc61c11ab352fd169067aef5a3a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.861ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} _{10}\left(a\ x\ b\right)}{\operatorname {d} x}}=a}" loading="lazy">

</td>

</td>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} _{11}\left(a\ x\ b\right)}{\operatorname {d} x}}=b}</annotation>

</semantics>

</math><img src="./3a8adda1f07e45abe8ae4b0cd820324e63b40989.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.629ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} _{11}\left(a\ x\ b\right)}{\operatorname {d} x}}=b}" loading="lazy">

</td></tr></tbody></table>

Similarly, for the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=x^{2}\ ,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ f(x)=x^{2}\ ,}</annotation>

</semantics>

</math><img src="./c7a998a3983a0d259e31face1a34a534d3ba2bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.708ex; height:3.176ex;" alt="{\displaystyle \ f(x)=x^{2}\ ,}" loading="lazy"> the derivative is

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} x^{2}}{\operatorname {d} x}}=x\otimes 1+1\otimes x}</annotation>

</semantics>

</math><img src="./4f296a232d98608f30491a2716923dd9e136938f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.504ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} x^{2}}{\operatorname {d} x}}=x\otimes 1+1\otimes x}" loading="lazy">

</td>

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {d} y={\frac {\operatorname {d} x^{2}}{\operatorname {d} x}}\circ \operatorname {d} x=x\ \operatorname {d} x+(\operatorname {d} x)\ x}</annotation>

</semantics>

</math><img src="./07a263f002a41488c40f59ee2d7bdea1d0f4b4c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.012ex; height:5.843ex;" alt="{\displaystyle \operatorname {d} y={\frac {\operatorname {d} x^{2}}{\operatorname {d} x}}\circ \operatorname {d} x=x\ \operatorname {d} x+(\operatorname {d} x)\ x}" loading="lazy">

</td></tr></tbody></table>

and the components are:

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} _{10}x^{2}}{\operatorname {d} x}}=x}</annotation>

</semantics>

</math><img src="./b438bf416adb318cf52fe5c1a5cdd6e3e90f1709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.204ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} _{10}x^{2}}{\operatorname {d} x}}=x}" loading="lazy">

</td>

</td>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} _{11}x^{2}}{\operatorname {d} x}}=1}</annotation>

</semantics>

</math><img src="./9e3102a67459156ba44282a41599968364ec7361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.037ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} _{11}x^{2}}{\operatorname {d} x}}=1}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} _{20}x^{2}}{\operatorname {d} x}}=1}</annotation>

</semantics>

</math><img src="./9554bde8e4bf1cad7d8f1b0b0ab155cc3c0ed474.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.037ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} _{20}x^{2}}{\operatorname {d} x}}=1}" loading="lazy">

</td>

</td>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} _{21}x^{2}}{\operatorname {d} x}}=x}</annotation>

</semantics>

</math><img src="./e92da5033397abb9ad60da0538bf79a1d041e20b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.204ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} _{21}x^{2}}{\operatorname {d} x}}=x}" loading="lazy">

</td></tr></tbody></table>

Finally, for the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f(x)=x^{-1}\ ,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d855cfdfa9118bea3988df48f11d7fb7ea49a4fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.987ex; height:3.176ex;" alt="{\displaystyle \ f(x)=x^{-1}\ ,}" loading="lazy"> the derivative is

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} x^{-1}}{\operatorname {d} x}}=-x^{-1}\otimes x^{-1}}</annotation>

</semantics>

</math><img src="./748d578948fc6c306e7ebf310f410fb2202ed0a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.25ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} x^{-1}}{\operatorname {d} x}}=-x^{-1}\otimes x^{-1}}" loading="lazy">

</td>

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {d} y={\frac {\operatorname {d} x^{-1}}{\operatorname {d} x}}\circ \operatorname {d} x=-x^{-1}(\operatorname {d} x)\ x^{-1}}</annotation>

</semantics>

</math><img src="./6e138c21348526fb766fe2396cbaa15274d12937.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.947ex; height:5.843ex;" alt="{\displaystyle \operatorname {d} y={\frac {\operatorname {d} x^{-1}}{\operatorname {d} x}}\circ \operatorname {d} x=-x^{-1}(\operatorname {d} x)\ x^{-1}}" loading="lazy">

</td></tr></tbody></table>

and the components are:

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} _{10}x^{-1}}{\operatorname {d} x}}=-x^{-1}}</annotation>

</semantics>

</math><img src="./93b29b5ab0f5ecbef5d9c467493d5483e15bbebb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.624ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} _{10}x^{-1}}{\operatorname {d} x}}=-x^{-1}}" loading="lazy">

</td>

</td>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {d} _{11}x^{-1}}{\operatorname {d} x}}=x^{-1}}</annotation>

</semantics>

</math><img src="./7e2864156a6ce799e6a7cb1d0e291b9be997d4e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.815ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {d} _{11}x^{-1}}{\operatorname {d} x}}=x^{-1}}" loading="lazy">

</td></tr></tbody></table>

<ul><li><a href="Cayley_transform" title="Cayley transform">Cayley transform</a></li>

<li><a href="Quaternionic_manifold" title="Quaternionic manifold">Quaternionic manifold</a></li></ul>

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

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFDeavours1973">Deavours (1973)</a> recalls a 1935 issue of <a href="Commentarii_Mathematici_Helvetici" title="Commentarii Mathematici Helvetici">Commentarii Mathematici Helvetici</a> where an alternative theory of "regular functions" was initiated by <a href="#CITEREFFueter1936">Fueter (1936)</a> through the idea of <a href="Morera's_theorem" title="Morera's theorem">Morera's theorem</a>: quaternion function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is "left regular at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">" when the integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> vanishes over any sufficiently small <a href="Hypersurface" title="Hypersurface">hypersurface</a> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">. Then the analogue of <a href="Liouville's_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a> holds: The only regular quaternion function with bounded norm in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} ^{4}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} ^{4}}</annotation>

</semantics>

</math><img src="./8e02756196353582bfdbb6b37fec4fb23e6996d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {E} ^{4}}" loading="lazy"> is a constant. One approach to construct regular functions is to use <a href="Power_series" title="Power series">power series</a> with real coefficients. Deavours also gives analogues for the <a href="Poisson_integral" class="mw-redirect" title="Poisson integral">Poisson integral</a>, the <a href="Cauchy_integral_formula" class="mw-redirect" title="Cauchy integral formula">Cauchy integral formula</a>, and the presentation of <a href="Maxwell%E2%80%99s_equations" class="mw-redirect" title="Maxwell’s equations">Maxwell’s equations</a> of electromagnetism with quaternion functions.

</li>

</ol></div></div>

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

<li id="cite_note-1"><a href="#cite_ref-1">^</a> (<a href="#CITEREFFueter1936">Fueter 1936</a>)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> (<a href="#CITEREFCayley1848">Cayley 1848</a>, especially page 198)

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> (<a href="#CITEREFHamilton1853">Hamilton 1853</a>, §287 pp. 273,4)

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFHamilton1866">Hamilton (1866)</a>, Chapter II, On differentials and developments of functions of quaternions, pp. 391–495

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFLaisant1881">Laisant (1881)</a>, Chapitre 5: Différentiation des Quaternions, pp. 104–117

</li>

</ol></div>

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

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFArnold1995" class="citation cs2"><a href="Vladimir_Arnold" title="Vladimir Arnold">Arnold, Vladimir</a> (1995), "The geometry of spherical curves and the algebra of quaternions", Russian Mathematical Surveys, 50 (1), translated by <a href="Ian_R._Porteous" title="Ian R. Porteous">Porteous, Ian R.</a>: 1–68, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1070%2FRM1995v050n01ABEH001662">10.1070/RM1995v050n01ABEH001662</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250897899">250897899</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0848.58005">0848.58005</a></cite></li>

<li><cite id="CITEREFCayley1848" class="citation cs2"><a href="Arthur_Cayley" title="Arthur Cayley">Cayley, Arthur</a> (1848), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1782007">"On the application of quaternions to the theory of rotation"</a>, <a href="London_and_Edinburgh_Philosophical_Magazine" class="mw-redirect" title="London and Edinburgh Philosophical Magazine">London and Edinburgh Philosophical Magazine</a>, Series 3, 33 (221): 196–200, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786444808645844">10.1080/14786444808645844</a></cite></li>

<li><cite id="CITEREFDeavours1973" class="citation cs2">Deavours, C.A. (1973), "The quaternion calculus", <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>, 80 (9), Washington, DC: Mathematical Association of America: 995–1008, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2318774">10.2307/2318774</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2318774">2318774</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0282.30040">0282.30040</a></cite></li>

<li><cite id="CITEREFDu_Val1964" class="citation cs2"><a href="Patrick_du_Val" title="Patrick du Val">Du Val, Patrick</a> (1964), Homographies, Quaternions and Rotations, Oxford Mathematical Monographs, Oxford: Clarendon Press, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0169108">0169108</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0128.15403">0128.15403</a></cite></li>

<li><cite id="CITEREFFueter1936" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Rudolf_Fueter" title="Rudolf Fueter">Fueter, Rudolf</a> (1936), "Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen", <a href="Commentarii_Mathematici_Helvetici" title="Commentarii Mathematici Helvetici">Commentarii Mathematici Helvetici</a> (in German), 8: 371–378, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01199562">10.1007/BF01199562</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121227604">121227604</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0014.16702">0014.16702</a></cite></li>

<li><cite id="CITEREFGentiliStoppatoStruppa2013" class="citation cs2">Gentili, Graziano; Stoppato, Caterina; Struppa, Daniele C. (2013), Regular Functions of a Quaternionic Variable, Berlin: Springer, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-33871-7">10.1007/978-3-642-33871-7</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-33870-0</bdi>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118710284">118710284</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1269.30001">1269.30001</a></cite></li>

<li><cite id="CITEREFGormley1947" class="citation cs2">Gormley, P.G. (1947), "Stereographic projection and the linear fractional group of transformations of quaternions", <a href="Proceedings_of_the_Royal_Irish_Academy%2C_Section_A" class="mw-redirect" title="Proceedings of the Royal Irish Academy, Section A">Proceedings of the Royal Irish Academy, Section A</a>, 51: 67–85, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/20488472">20488472</a></cite></li>

<li><cite id="CITEREFGürlebeckSprößig1990" class="citation cs2">Gürlebeck, Klaus; Sprößig, Wolfgang (1990), Quaternionic analysis and elliptic boundary value problems, Basel: Birkhäuser, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-2382-0</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0850.35001">0850.35001</a></cite></li>

<li><cite id="CITEREFJohn_C.Holladay1957" class="citation cs2">John C.Holladay (1957), <a rel="nofollow" class="external text" href="http://www.ams.org/journals/proc/1957-008-04/S0002-9939-1957-0087047-7/S0002-9939-1957-0087047-7.pdf">"The Stone–Weierstrass theorem for quaternions"</a> (PDF), Proc. Amer. Math. Soc., 8: 656, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1957-0087047-7">10.1090/S0002-9939-1957-0087047-7</a></cite>.</li>

<li><cite id="CITEREFHamilton1853" class="citation cs2"><a href="William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton, William Rowan</a> (1853), Lectures on Quaternions, Dublin: Hodges and Smith, <a href="OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL23416635M">23416635M</a></cite></li>

<li><cite id="CITEREFHamilton1866" class="citation cs2"><a href="William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton, William Rowan</a> (1866), <a href="William_Edwin_Hamilton" title="William Edwin Hamilton">Hamilton, William Edwin</a> (ed.), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b2stAAAAYAAJ">Elements of Quaternions</a>, London: Longmans, Green, & Company, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1204.01046">1204.01046</a></cite></li>

<li><cite id="CITEREFJoly1903" class="citation cs2"><a href="Charles_Jasper_Joly" title="Charles Jasper Joly">Joly, Charles Jasper</a> (1903), "Quaternions and projective geometry", <a href="Philosophical_Transactions_of_the_Royal_Society_of_London" class="mw-redirect" title="Philosophical Transactions of the Royal Society of London">Philosophical Transactions of the Royal Society of London</a>, 201 (331–345): 223–327, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1903RSPTA.201..223J">1903RSPTA.201..223J</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1903.0018">10.1098/rsta.1903.0018</a>, <a href="JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:34.0092.01">34.0092.01</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/90902">90902</a></cite></li>

<li><cite id="CITEREFLaisant1881" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Charles-Ange_Laisant" title="Charles-Ange Laisant">Laisant, Charles-Ange</a> (1881), <a rel="nofollow" class="external text" href="https://archive.org/details/introductionlam01laisgoog">Introduction à la Méthode des Quaternions</a> (in French), Paris: Gauthier-Villars, <a href="JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:13.0524.02">13.0524.02</a></cite></li>

<li><cite id="CITEREFPorter1998" class="citation cs2">Porter, R. Michael (1998), <a rel="nofollow" class="external text" href="https://www.ams.org/journals/ecgd/1998-02-06/S1088-4173-98-00032-0/S1088-4173-98-00032-0.pdf">"Möbius invariant quaternion geometry"</a> (PDF), Conformal Geometry and Dynamics, 2 (6): 89–196, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS1088-4173-98-00032-0">10.1090/S1088-4173-98-00032-0</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0910.53005">0910.53005</a></cite></li>

<li><cite id="CITEREFSudbery1979" class="citation cs2">Sudbery, A. (1979), "Quaternionic analysis", <a href="Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society" title="Mathematical Proceedings of the Cambridge Philosophical Society">Mathematical Proceedings of the Cambridge Philosophical Society</a>, 85 (2): 199–225, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979MPCPS..85..199S">1979MPCPS..85..199S</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100055638">10.1017/S0305004100055638</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F101933">10338.dmlcz/101933</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7606387">7606387</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0399.30038">0399.30038</a></cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Analysis_in_topological_vector_spaces159" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Analysis_in_topological_vector_spaces159" style="font-size:114%;margin:0 4em"><a href="Mathematical_analysis" title="Mathematical analysis">Analysis</a> in <a href="Topological_vector_space" title="Topological vector space">topological vector spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical Wiener space</a></li></ul></li>

<li><a href="Bochner_space" title="Bochner space">Bochner space</a></li>

<li><a href="Convex_series" title="Convex series">Convex series</a></li>

<li><a href="Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li>

<li><a href="Infinite-dimensional_vector_function" title="Infinite-dimensional vector function">Infinite-dimensional vector function</a></li>

<li><a href="Matrix_calculus" title="Matrix calculus">Matrix calculus</a></li>

<li><a href="Vector_calculus" title="Vector calculus">Vector calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Derivative" title="Derivative">Derivatives</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Differentiable_vector-valued_functions_from_Euclidean_space" title="Differentiable vector-valued functions from Euclidean space">Differentiable vector-valued functions from Euclidean space</a></li>

<li><a href="Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li>

<li><a href="Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a>

<ul><li><a href="Total_derivative" title="Total derivative">Total</a></li></ul></li>

<li><a href="Functional_derivative" title="Functional derivative">Functional derivative</a></li>

<li><a href="Gateaux_derivative" title="Gateaux derivative">Gateaux derivative</a>

<ul><li><a href="Directional_derivative" title="Directional derivative">Directional</a></li></ul></li>

<li><a href="Generalizations_of_the_derivative" title="Generalizations of the derivative">Generalizations of the derivative</a></li>

<li><a href="Hadamard_derivative" title="Hadamard derivative">Hadamard derivative</a></li>

<li><a href="Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">Holomorphic</a></li>

<li><a href="Quasi-derivative" title="Quasi-derivative">Quasi-derivative</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Measurability</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Besov_measure" title="Besov measure">Besov measure</a></li>

<li><a href="Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a>

<ul><li><a href="Canonical_Gaussian_cylinder_set_measure" class="mw-redirect" title="Canonical Gaussian cylinder set measure">Canonical Gaussian</a></li>

<li><a href="Classical_Wiener_measure" class="mw-redirect" title="Classical Wiener measure">Classical Wiener measure</a></li></ul></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure</a> like <a href="Set_function" title="Set function">set functions</a>

<ul><li><a href="Gaussian_measure#Infinite-dimensional_spaces" title="Gaussian measure">infinite-dimensional Gaussian measure</a></li>

<li><a href="Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li>

<li><a href="Vector_measure" title="Vector measure">Vector</a></li></ul></li>

<li><a href="Bochner_measurable_function" title="Bochner measurable function">Bochner</a> / <a href="Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="Strongly_measurable_function" title="Strongly measurable function">Strongly</a> <a href="Measurable_function" title="Measurable function">measurable function</a></li>

<li><a href="Radonifying_function" title="Radonifying function">Radonifying function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integral" title="Integral">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bochner_integral" title="Bochner integral">Bochner</a></li>

<li><a href="Direct_integral" title="Direct integral">Direct integral</a></li>

<li><a href="Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li>

<li><a href="Pettis_integral" title="Pettis integral">Gelfand–Pettis/Weak</a></li>

<li><a href="Regulated_integral" title="Regulated integral">Regulated</a></li>

<li><a href="Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cameron%E2%80%93Martin_theorem" title="Cameron–Martin theorem">Cameron–Martin theorem</a></li>

<li><a href="Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a>

<ul><li><a href="Nash%E2%80%93Moser_theorem" title="Nash–Moser theorem">Nash–Moser theorem</a></li></ul></li>

<li><a href="Feldman%E2%80%93H%C3%A1jek_theorem" title="Feldman–Hájek theorem">Feldman–Hájek theorem</a></li>

<li><a href="Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">No infinite-dimensional Lebesgue measure</a></li>

<li><a href="Sazonov's_theorem" title="Sazonov's theorem">Sazonov's theorem</a></li>

<li><a href="Structure_theorem_for_Gaussian_measures" title="Structure theorem for Gaussian measures">Structure theorem for Gaussian measures</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Crinkled_arc" title="Crinkled arc">Crinkled arc</a></li>

<li><a href="Covariance_operator" title="Covariance operator">Covariance operator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Functional_calculus" title="Functional calculus">Functional calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li>

<li><a href="Continuous_functional_calculus" title="Continuous functional calculus">Continuous functional calculus</a></li>

<li><a href="Holomorphic_functional_calculus" title="Holomorphic functional calculus">Holomorphic functional calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_manifold" title="Banach manifold">Banach manifold</a> (<a href="Banach_bundle" title="Banach bundle">bundle</a>)</li>

<li><a href="Convenient_vector_space" title="Convenient vector space">Convenient vector space</a></li>

<li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li>

<li><a href="Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-02-26" href="https://en.wikipedia.org/wiki/?title=Quaternionic_analysis&oldid=1277731349">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>