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<title>Rational function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Rational function</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For the use in automata theory, see <a href="Finite-state_transducer" title="Finite-state transducer">Finite-state transducer</a>. For the use in monoid theory, see <a href="Rational_function_(monoid)" class="mw-redirect" title="Rational function (monoid)">Rational function (monoid)</a>.</div>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>rational function</b> is any <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that can be defined by a <b>rational fraction</b>, which is an <a href="Algebraic_fraction" title="Algebraic fraction">algebraic fraction</a> such that both the <a href="Numerator" class="mw-redirect" title="Numerator">numerator</a> and the <a href="Denominator" class="mw-redirect" title="Denominator">denominator</a> are <a href="Polynomial" title="Polynomial">polynomials</a>. The <a href="Coefficient" title="Coefficient">coefficients</a> of the polynomials need not be <a href="Rational_number" title="Rational number">rational numbers</a>; they may be taken in any <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml mvar" style="font-style:italic;">K</span>. In this case, one speaks of a rational function and a rational fraction <i>over <span class="texhtml mvar" style="font-style:italic;">K</span></i>. The values of the <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> may be taken in any field <span class="texhtml mvar" style="font-style:italic;">L</span> containing <span class="texhtml mvar" style="font-style:italic;">K</span>. Then the <a href="Domain_(function)" class="mw-redirect" title="Domain (function)">domain</a> of the function is the set of the values of the variables for which the denominator is not zero, and the <a href="Codomain" title="Codomain">codomain</a> is <span class="texhtml mvar" style="font-style:italic;">L</span>.

</p><p>The set of rational functions over a field <span class="texhtml mvar" style="font-style:italic;">K</span> is a field, the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> of the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of the <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> over <span class="texhtml mvar" style="font-style:italic;">K</span>.

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

<p>A function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is called a rational function if it can be written in the form<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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

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</math></span><img src="./fd06ca086165f81ca3d20e652534036333c53050.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.329ex; height:6.509ex;" alt="{\displaystyle f(x)={\frac {P(x)}{Q(x)}}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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

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</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> are <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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

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</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> is not the <a href="Zero_function" class="mw-redirect" title="Zero function">zero function</a>. The <a href="Domain_of_a_function" title="Domain of a function">domain</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is the set of all values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.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 x}" loading="lazy"></span> for which the denominator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

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

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</math></span><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"></span> is not zero.

</p><p>However, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle P}">

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

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</math></span><img src="./c2ca15ed6f7aeb18a38d8be3cf18abcd4b617666.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle \textstyle P}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle Q}">

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

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</math></span><img src="./90e059b946d49fffd482d23a4df6ad33fc179146.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle \textstyle Q}" loading="lazy"></span> have a non-constant <a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">polynomial greatest common divisor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle R}">

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

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</math></span><img src="./41b44af7e47b69695c83d364488950da2e9cda5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle \textstyle R}" loading="lazy"></span>, then setting <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle P=P_{1}R}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle P=P_{1}R}</annotation>

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</math></span><img src="./85334b5fc911ec820d1e96a4b1f8836909d9db1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.154ex; height:2.509ex;" alt="{\displaystyle \textstyle P=P_{1}R}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle Q=Q_{1}R}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle Q=Q_{1}R}</annotation>

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</math></span><img src="./39de1f0003fbb3f85c5198807f940c5a4b61cb83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.593ex; height:2.509ex;" alt="{\displaystyle \textstyle Q=Q_{1}R}" loading="lazy"></span> produces a rational function

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<annotation encoding="application/x-tex">{\displaystyle f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},}</annotation>

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</math></span><img src="./24f4955cad8e5e7cb7f01577be3016edbec4087b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.945ex; height:6.509ex;" alt="{\displaystyle f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},}" loading="lazy"></span></dd></dl>

<p>which may have a larger domain than <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>, and is equal to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> on the domain of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">

</mstyle>

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

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</math></span><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"></span> It is a common usage to identify <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}}">

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

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</math></span><img src="./50dfd257a51e037112c917f8a9e47c9c053466df.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_{1}}" loading="lazy"></span>, that is to extend "by continuity" the domain of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> to that of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}.}">

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

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</math></span><img src="./15589063e81f3e5fa2699321756b83cab936bb91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.509ex;" alt="{\displaystyle f_{1}.}" loading="lazy"></span> Indeed, one can define a rational fraction as an <a href="Equivalence_class" title="Equivalence class">equivalence class</a> of fractions of polynomials, where two fractions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {A(x)}{B(x)}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {A(x)}{B(x)}}}</annotation>

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</math></span><img src="./ff8a100d884ab5c80e1bee7cfa74f5e41ac796ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.303ex; height:4.843ex;" alt="{\displaystyle \textstyle {\frac {A(x)}{B(x)}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {C(x)}{D(x)}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {C(x)}{D(x)}}}</annotation>

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</math></span><img src="./1b28972ad1509dc51d43bc66e3d83fe4a26516dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.416ex; height:4.843ex;" alt="{\displaystyle \textstyle {\frac {C(x)}{D(x)}}}" loading="lazy"></span> are considered equivalent if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x)D(x)=B(x)C(x)}">

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<annotation encoding="application/x-tex">{\displaystyle A(x)D(x)=B(x)C(x)}</annotation>

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</math></span><img src="./41ee2f8207faa8b763834526fe537212705871f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.852ex; height:2.843ex;" alt="{\displaystyle A(x)D(x)=B(x)C(x)}" loading="lazy"></span>. In this case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {P(x)}{Q(x)}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {P(x)}{Q(x)}}}</annotation>

</semantics>

</math></span><img src="./140939176c5bff86497ef646402e6597c0ac66be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.356ex; height:4.843ex;" alt="{\displaystyle \textstyle {\frac {P(x)}{Q(x)}}}" loading="lazy"></span> is equivalent to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {P_{1}(x)}{Q_{1}(x)}}.}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {P_{1}(x)}{Q_{1}(x)}}.}</annotation>

</semantics>

</math></span><img src="./6d9dc1909df5fce7a51ba6f9fb775a28e550b049.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.834ex; height:4.843ex;" alt="{\displaystyle \textstyle {\frac {P_{1}(x)}{Q_{1}(x)}}.}" loading="lazy"></span>

</p><p>A <b>proper rational function</b> is a rational function in which the <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./89833156eff2c51bfb8750db3306a0544ce34e14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" loading="lazy"></span> is less than the degree of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"></span> and both are <a href="Real_polynomial" class="mw-redirect" title="Real polynomial">real polynomials</a>, named by analogy to a <a href="Fraction#Proper_and_improper_fractions" title="Fraction">proper fraction</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} .}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./869719f08f506bf866043442858fb3da1d4b4b5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} .}" loading="lazy"></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Complex_rational_functions">Complex rational functions</h3></div>

<p>In <a href="Complex_analysis" title="Complex analysis">complex analysis</a>, a rational function

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {P(z)}{Q(z)}}}</annotation>

</semantics>

</math></span><img src="./9df4b32242abb4c8d015bdb82d2e49a8d6b37fb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.846ex; height:6.509ex;" alt="{\displaystyle f(z)={\frac {P(z)}{Q(z)}}}" loading="lazy"></span></dd></dl>

<p>is the ratio of two polynomials with complex coefficients, where <span class="texhtml"><i>Q</i></span> is not the zero polynomial and <span class="texhtml"><i>P</i></span> and <span class="texhtml"><i>Q</i></span> have no common factor (this avoids <span class="texhtml"><i>f</i></span> taking the indeterminate value 0/0).

</p><p>The domain of <span class="texhtml mvar" style="font-style:italic;">f</span> is the set of complex numbers such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(z)\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ad338f72ba79b2c42829a97ba9b373322da3034f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.997ex; height:2.843ex;" alt="{\displaystyle Q(z)\neq 0}" loading="lazy"></span>.

Every rational function can be naturally extended to a function whose domain and range are the whole <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>, i.e., a <a href="Rational_mapping" title="Rational mapping">rational mapping</a>. Iteration of rational functions on the Riemann sphere forms a <a href="Discrete_dynamical_system" class="mw-redirect" title="Discrete dynamical system">discrete dynamical system</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>A complex rational function with degree one is a <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a>.

</p><p>Rational functions are representative examples of <a href="Meromorphic_function" title="Meromorphic function">meromorphic functions</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<li class="gallerycaption"><a href="Julia_set" title="Julia set">Julia sets</a> for rational maps</li>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{az^{5}+z^{3}+bz}}}</annotation>

</semantics>

</math></span><img src="./ef5648ff6dc738902ce4121044f8b608f840b34b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.122ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{az^{5}+z^{3}+bz}}}" loading="lazy"></span></div>

</li>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{z^{3}+z(-3-3i)}}}</annotation>

</semantics>

</math></span><img src="./1fe6e1be22e80ebc9471a864dd973236ca3101e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.494ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{z^{3}+z(-3-3i)}}}" loading="lazy"></span></div>

</li>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {z^{2}-0.2+0.7i}{z^{2}+0.917}}}</annotation>

</semantics>

</math></span><img src="./aa6613e39fd925835a904325646e8b2701eb1d8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.407ex; height:6.176ex;" alt="{\displaystyle {\frac {z^{2}-0.2+0.7i}{z^{2}+0.917}}}" loading="lazy"></span></div>

</li>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {z^{2}}{z^{9}-z+0.025}}}</annotation>

</semantics>

</math></span><img src="./da67d5026810e3dfefb57d5d89742d6eb2488aa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.046ex; height:6.176ex;" alt="{\displaystyle {\frac {z^{2}}{z^{9}-z+0.025}}}" loading="lazy"></span></div>

</li>

</ul>

<div class="mw-heading mw-heading3"><h3 id="Degree">Degree</h3></div>

<p>There are several non equivalent definitions of the degree of a rational function.

</p><p>Most commonly, the <i>degree</i> of a rational function is the maximum of the <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degrees</a> of its constituent polynomials <span class="texhtml"><i>P</i></span> and <span class="texhtml"><i>Q</i></span>, when the fraction is reduced to <a href="Lowest_terms" class="mw-redirect" title="Lowest terms">lowest terms</a>. If the degree of <span class="texhtml"><i>f</i></span> is <span class="texhtml"><i>d</i></span>, then the equation

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./14a3ed715624604b2731c41f44ab34e0f022ada5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.326ex; height:2.843ex;" alt="{\displaystyle f(z)=w\,}" loading="lazy"></span></dd></dl>

<p>has <span class="texhtml"><i>d</i></span> distinct solutions in <span class="texhtml"><i>z</i></span> except for certain values of <span class="texhtml"><i>w</i></span>, called <i>critical values</i>, where two or more solutions coincide or where some solution is rejected <a href="Point_at_infinity" title="Point at infinity">at infinity</a> (that is, when the degree of the equation decreases after having <a href="Clearing_denominators" title="Clearing denominators">cleared the denominator</a>).

</p><p>The <a href="Degree_of_an_algebraic_variety" title="Degree of an algebraic variety">degree</a> of the <a href="Graph_of_a_function" title="Graph of a function">graph</a> of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.

</p><p>In some contexts, such as in <a href="Asymptotic_analysis" title="Asymptotic analysis">asymptotic analysis</a>, the <i>degree</i> of a rational function is the difference between the degrees of the numerator and the denominator.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: §13.6.1">: §13.6.1 </span></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Chapter IV">: Chapter IV </span></sup>

</p><p>In <a href="Network_synthesis" title="Network synthesis">network synthesis</a> and <a href="Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">network analysis</a>, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a <b><style data-mw-deduplicate="TemplateStyles:r1238216509">

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

.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}

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

</style><span class="vanchor"><span class="vanchor-text">biquadratic function</span></span></b>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

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

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.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:304px;max-width:304px"><div class="trow"><div class="theader">Examples of rational functions</div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Rational function of degree 3, with a graph of <a href="Degree_of_an_algebraic_variety" title="Degree of an algebraic variety">degree</a> 3: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\frac {x^{3}-2x}{2(x^{2}-5)}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y={\frac {x^{3}-2x}{2(x^{2}-5)}}}</annotation>

</semantics>

</math></span><img src="./42b052f9a5a6814546910759e922317c909686c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.449ex; height:6.509ex;" alt="{\displaystyle y={\frac {x^{3}-2x}{2(x^{2}-5)}}}" loading="lazy"></span></div></div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Rational function of degree 2, with a graph of <a href="Degree_of_an_algebraic_variety" title="Degree of an algebraic variety">degree</a> 3: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\frac {x^{2}-3x-2}{x^{2}-4}}}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y={\frac {x^{2}-3x-2}{x^{2}-4}}}</annotation>

</semantics>

</math></span><img src="./f34422d5e504da2510ad790b946b2af8d5515976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.809ex; height:6.176ex;" alt="{\displaystyle y={\frac {x^{2}-3x-2}{x^{2}-4}}}" loading="lazy"></span></div></div></div></div></div>

<p>The rational function

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}}</annotation>

</semantics>

</math></span><img src="./3b61c0c51715402b52c938bf4f747e76cbf0d409.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.711ex; height:6.509ex;" alt="{\displaystyle f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}}" loading="lazy"></span></dd></dl>

<p>is not defined at

</p>

<msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}.}</annotation>

</semantics>

</math></span><img src="./75934be19be8dfaa9fcef26704ded9e929e572d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.24ex; height:3.009ex;" alt="{\displaystyle x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}.}" loading="lazy"></span></dd></dl>

<p>It is asymptotic to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {x}{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e268b5e3776a292d839263c29b18be82f488dfa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.776ex; height:3.176ex;" alt="{\displaystyle {\tfrac {x}{2}}}" loading="lazy"></span> as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \infty .}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./077f3645400fedfac226358e8dc3ead4949b1fc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.914ex; height:1.843ex;" alt="{\displaystyle x\to \infty .}" loading="lazy"></span>

</p><p>The rational function

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./01453176f8b8e783b9bbf9e7fb9336c5f1fe8658.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.739ex; height:6.176ex;" alt="{\displaystyle f(x)={\frac {x^{2}+2}{x^{2}+1}}}" loading="lazy"></span></dd></dl>

<p>is defined for all <a href="Real_number" title="Real number">real numbers</a>, but not for all <a href="Complex_number" title="Complex number">complex numbers</a>, since if <i>x</i> were a square root of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"></span> (i.e. the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a> or its negative), then formal evaluation would lead to division by zero:

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}},}</annotation>

</semantics>

</math></span><img src="./f262128ed321b60a4a06f09b515daa385e10f237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.336ex; height:6.176ex;" alt="{\displaystyle f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}},}" loading="lazy"></span></dd></dl>

<p>which is undefined.

</p><p>A <a href="Constant_function" title="Constant function">constant function</a> such as <span class="texhtml"><i>f</i>(<i>x</i>) = π</span> is a rational function since constants are polynomials. The function itself is rational, even though the <a href="Value_(mathematics)" title="Value (mathematics)">value</a> of <span class="texhtml"><i>f</i>(<i>x</i>)</span> is irrational for all <span class="texhtml mvar" style="font-style:italic;">x</span>.

</p><p>Every <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=P(x)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a975d2c42ffc827b8e33dbe7da4e1b1641f045f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.4ex; height:2.843ex;" alt="{\displaystyle f(x)=P(x)}" loading="lazy"></span> is a rational function with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=1.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)=1.}</annotation>

</semantics>

</math></span><img src="./f964e4f8242d245e0037f068a0b87205307e572c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.885ex; height:2.843ex;" alt="{\displaystyle Q(x)=1.}" loading="lazy"></span> A function that cannot be written in this form, such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sin(x),}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./aa19dc0b6340a89fa50e0bbdd1ef381650f9651f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.157ex; height:2.843ex;" alt="{\displaystyle f(x)=\sin(x),}" loading="lazy"></span> is not a rational function. However, the adjective "irrational" is <b>not</b> generally used for functions.

</p><p>Every <a href="Laurent_polynomial" title="Laurent polynomial">Laurent polynomial</a> can be written as a rational function while the converse is not necessarily true, i.e., the ring of Laurent polynomials is a <a href="Subring" title="Subring">subring</a> of the rational functions.

</p><p>The rational function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\tfrac {x}{x}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a5d17f698541dedcab8a952750682807dd9fd758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.292ex; height:3.009ex;" alt="{\displaystyle f(x)={\tfrac {x}{x}}}" loading="lazy"></span> is equal to 1 for all <i>x</i> except 0, where there is a <a href="Removable_singularity" title="Removable singularity">removable singularity</a>. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since <i>x</i>/<i>x</i> is equivalent to 1/1.

</p>

<div class="mw-heading mw-heading2"><h2 id="Taylor_series">Taylor series</h2></div>

<p>The coefficients of a <a href="Taylor_series" title="Taylor series">Taylor series</a> of any rational function satisfy a <a href="Recurrence_relation" title="Recurrence relation">linear recurrence relation</a>, which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collecting <a href="Like_terms" title="Like terms">like terms</a> after clearing the denominator.

</p><p>For example,

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x^{2}-x+2}}=\sum _{k=0}^{\infty }a_{k}x^{k}.}</annotation>

</semantics>

</math></span><img src="./4d2e1cf4e8a319f0427a06e299cd150c43caadd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.617ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{x^{2}-x+2}}=\sum _{k=0}^{\infty }a_{k}x^{k}.}" loading="lazy"></span></dd></dl>

<p>Multiplying through by the denominator and distributing,

</p>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=(x^{2}-x+2)\sum _{k=0}^{\infty }a_{k}x^{k}}</annotation>

</semantics>

</math></span><img src="./e7ca5cd8802fec2ab0fdff09da7ece43c1b6b8c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.493ex; height:7.009ex;" alt="{\displaystyle 1=(x^{2}-x+2)\sum _{k=0}^{\infty }a_{k}x^{k}}" loading="lazy"></span></dd></dl>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=\sum _{k=0}^{\infty }a_{k}x^{k+2}-\sum _{k=0}^{\infty }a_{k}x^{k+1}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}</annotation>

</semantics>

</math></span><img src="./6c97f845640b6640c38c5a3f3784471787252934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.776ex; height:7.009ex;" alt="{\displaystyle 1=\sum _{k=0}^{\infty }a_{k}x^{k+2}-\sum _{k=0}^{\infty }a_{k}x^{k+1}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}" loading="lazy"></span></dd></dl>

<p>After adjusting the indices of the sums to get the same powers of <i>x</i>, we get

</p>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=\sum _{k=2}^{\infty }a_{k-2}x^{k}-\sum _{k=1}^{\infty }a_{k-1}x^{k}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}</annotation>

</semantics>

</math></span><img src="./92222ca4856f337fdab13c88684d50fe113fdb01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.776ex; height:7.009ex;" alt="{\displaystyle 1=\sum _{k=2}^{\infty }a_{k-2}x^{k}-\sum _{k=1}^{\infty }a_{k-1}x^{k}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}" loading="lazy"></span></dd></dl>

<p>Combining like terms gives

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=2a_{0}+(2a_{1}-a_{0})x+\sum _{k=2}^{\infty }(a_{k-2}-a_{k-1}+2a_{k})x^{k}.}</annotation>

</semantics>

</math></span><img src="./fcb5b3ae5de2b21263de25ac4bf2903e9de3a4a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.328ex; height:6.843ex;" alt="{\displaystyle 1=2a_{0}+(2a_{1}-a_{0})x+\sum _{k=2}^{\infty }(a_{k-2}-a_{k-1}+2a_{k})x^{k}.}" loading="lazy"></span></dd></dl>

<p>Since this holds true for all <i>x</i> in the <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> of the original Taylor series, we can compute as follows. Since the <a href="Constant_term" title="Constant term">constant term</a> on the left must equal the constant term on the right it follows that

</p>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {1}{2}}.}</annotation>

</semantics>

</math></span><img src="./b3db42147015b301d3ae6116de4b6f5f813c2d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.028ex; height:5.176ex;" alt="{\displaystyle a_{0}={\frac {1}{2}}.}" loading="lazy"></span></dd></dl>

<p>Then, since there are no powers of <i>x</i> on the left, all of the <a href="Coefficient" title="Coefficient">coefficients</a> on the right must be zero, from which it follows that

</p>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1}={\frac {1}{4}}}</annotation>

</semantics>

</math></span><img src="./8bd3ccfbe3e22ad66a75422f3d1cfce09e9ff4a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.381ex; height:5.176ex;" alt="{\displaystyle a_{1}={\frac {1}{4}}}" loading="lazy"></span></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{k}={\frac {1}{2}}(a_{k-1}-a_{k-2})\quad {\text{for}}\ k\geq 2.}</annotation>

</semantics>

</math></span><img src="./1b9100d439b80193b57419716c9cf45549aa9a6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.712ex; height:5.176ex;" alt="{\displaystyle a_{k}={\frac {1}{2}}(a_{k-1}-a_{k-2})\quad {\text{for}}\ k\geq 2.}" loading="lazy"></span></dd></dl>

<p>Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using <a href="Partial_fraction" class="mw-redirect" title="Partial fraction">partial fraction decomposition</a> we can write any proper rational function as a sum of factors of the form <span class="nowrap">1 / (<i>ax</i> + <i>b</i>)</span> and expand these as <a href="Geometric_series" title="Geometric series">geometric series</a>, giving an explicit formula for the Taylor coefficients; this is the method of <a href="Generating_functions" class="mw-redirect" title="Generating functions">generating functions</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Abstract_algebra">Abstract algebra</h2></div>

<p>In <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a> the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any <a href="Field_(mathematics)" title="Field (mathematics)">field</a>. In this setting, given a field <i>F</i> and some indeterminate <i>X</i>, a <b>rational expression</b> (also known as a <b>rational fraction</b> or, in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <b>rational function</b>) is any element of the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> of the <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> <i>F</i>[<i>X</i>]. Any rational expression can be written as the quotient of two polynomials <i>P</i>/<i>Q</i> with <i>Q</i> ≠ 0, although this representation isn't unique. <i>P</i>/<i>Q</i> is equivalent to <i>R</i>/<i>S</i>, for polynomials <i>P</i>, <i>Q</i>, <i>R</i>, and <i>S</i>, when <i>PS</i> = <i>QR</i>. However, since <i>F</i>[<i>X</i>] is a <a href="Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a>, there is a <a href="Irreducible_fraction" title="Irreducible fraction">unique representation</a> for any rational expression <i>P</i>/<i>Q</i> with <i>P</i> and <i>Q</i> polynomials of lowest degree and <i>Q</i> chosen to be <a href="Monic_polynomial" title="Monic polynomial">monic</a>. This is similar to how a <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> of integers can always be written uniquely in lowest terms by canceling out common factors.

</p><p>The field of rational expressions is denoted <i>F</i>(<i>X</i>). This field is said to be generated (as a field) over <i>F</i> by (a <a href="Transcendental_element" class="mw-redirect" title="Transcendental element">transcendental element</a>) <i>X</i>, because <i>F</i>(<i>X</i>) does not contain any proper subfield containing both <i>F</i> and the element <i>X</i>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Notion_of_a_rational_function_on_an_algebraic_variety">Notion of a rational function on an algebraic variety</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">Function field of an algebraic variety</a></div>

<p>Like <a href="Polynomial_ring#The_polynomial_ring_in_several_variables" title="Polynomial ring">polynomials</a>, rational expressions can also be generalized to <i>n</i> indeterminates <i>X</i><sub>1</sub>,..., <i>X</i><sub><i>n</i></sub>, by taking the field of fractions of <i>F</i>[<i>X</i><sub>1</sub>,..., <i>X</i><sub><i>n</i></sub>], which is denoted by <i>F</i>(<i>X</i><sub>1</sub>,..., <i>X</i><sub><i>n</i></sub>).

</p><p>An extended version of the abstract idea of rational function is used in algebraic geometry. There the <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field of an algebraic variety</a> <i>V</i> is formed as the field of fractions of the <a href="Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a> of <i>V</i> (more accurately said, of a <a href="Zariski_topology" title="Zariski topology">Zariski</a>-<a href="Dense_subset" class="mw-redirect" title="Dense subset">dense</a> affine open set in <i>V</i>). Its elements <i>f</i> are considered as regular functions in the sense of algebraic geometry on non-empty open sets <i>U</i>, and also may be seen as morphisms to the <a href="Projective_line" title="Projective line">projective line</a>.

</p>

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

<p>Rational functions are used in <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a> for <a href="Interpolation" title="Interpolation">interpolation</a> and <a href="Approximation" title="Approximation">approximation</a> of functions, for example the <a href="Pad%C3%A9_approximant" title="Padé approximant">Padé approximants</a> introduced by <a href="Henri_Pad%C3%A9" title="Henri Padé">Henri Padé</a>. Approximations in terms of rational functions are well suited for <a href="Computer_algebra_system" title="Computer algebra system">computer algebra systems</a> and other numerical <a href="Software" title="Software">software</a>. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials.

</p><p>Rational functions are used to approximate or model more complex equations in science and engineering including <a href="Field_(physics)" title="Field (physics)">fields</a> and <a href="Force" title="Force">forces</a> in physics, <a href="Spectroscopy" title="Spectroscopy">spectroscopy</a> in analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo, <a href="Wave_function" title="Wave function">wave functions</a> for atoms and molecules, optics and photography to improve image resolution, and acoustics and sound.

</p><p>In <a href="Signal_processing" title="Signal processing">signal processing</a>, the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> (for continuous systems) or the <a href="Z-transform" title="Z-transform">z-transform</a> (for discrete-time systems) of the <a href="Impulse_response" title="Impulse response">impulse response</a> of commonly used <a href="Linear_time-invariant_system" title="Linear time-invariant system">linear time-invariant systems</a> (filters) with <a href="Infinite_impulse_response" title="Infinite impulse response">infinite impulse response</a> are rational functions over complex numbers.

</p>

<ul><li><a href="Partial_fraction_decomposition" title="Partial fraction decomposition">Partial fraction decomposition</a></li>

<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a></li>

<li><a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">Function field of an algebraic variety</a></li>

<li><a href="Algebraic_fraction" title="Algebraic fraction">Algebraic fractions</a> – a generalization of rational functions that allows taking integer roots</li></ul>

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

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</style><cite id="CITEREFRudin1987" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1987). <i>Real and Complex Analysis</i>. New York, NY: McGraw-Hill Education. p. 267. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-100276-9</bdi>.</cite></span>

</li>

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</style><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><cite id="CITEREFCorlessFrazho2003" class="citation book cs1">Corless, Martin J.; Frazho, Art (2003). <i>Linear Systems and Control</i>. CRC Press. p. 163. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0203911377</bdi>.</cite></li><li><cite id="CITEREFPownall1983" class="citation book cs1">Pownall, Malcolm W. (1983). <i>Functions and Graphs: Calculus Preparatory Mathematics</i>. Prentice-Hall. p. 203. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0133323048</bdi>.</cite></li></ul></div></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFBlanchard1984" class="citation journal cs1">Blanchard, Paul (1984). <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-11/issue-1/Complex-analytic-dynamics-on-the-Riemann-sphere/bams/1183551835.full">"Complex analytic dynamics on the Riemann sphere"</a>. <i>Bulletin of the American Mathematical Society</i>. <b>11</b> (1): <span class="nowrap">85–</span>141. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1984-15240-6">10.1090/S0273-0979-1984-15240-6</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0273-0979">0273-0979</a>.</cite> p. 87</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFAblowitzFokas2003" class="citation book cs1"><a href="Mark_Ablowitz" class="mw-redirect" title="Mark Ablowitz">Ablowitz, Mark J.</a>; <a href="Athanassios_Fokas" title="Athanassios Fokas">Fokas, Athanassios S.</a> (2003). <i>Complex Variables</i>. Cambridge University Press. p. 150. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-53429-1</bdi>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFBourles2010" class="citation book cs1">Bourles, Henri (2010). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9781118619988"><i>Linear Systems</i></a>. Wiley. p. 515. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781118619988">10.1002/9781118619988</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84821-162-9</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">5 November</span> 2022</span>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFBourbaki1990" class="citation book cs1"><a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, N.</a> (1990). <i>Algebra II</i>. Springer. p. A.IV.20. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-19375-8</bdi>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFGlisson2011" class="citation book cs1">Glisson, Tildon H. (2011). <i>Introduction to Circuit Analysis and Design</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-9048194438</bdi>.</cite></span>

</li>

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

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Rational_function">"Rational function"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007), <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html?pg=124">"Section 3.4. Rational Function Interpolation and Extrapolation"</a>, <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.), Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rational_functions">Dynamic visualization of rational functions with JSXGraph</a></li></ul>

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</style><div id="Function330" style="font-size:114%;margin:0 4em"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_the_function_concept" title="History of the function concept">History</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul>

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</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="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

<li><a href="Linear_map" title="Linear map">Linear</a></li>

<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</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="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>

<li><a href="Function_space" title="Function space">Space</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

<li><a href="Morphism" title="Morphism">Morphism</a></li>

<li><a href="Functor" title="Functor">Functor</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-06-24" href="https://en.wikipedia.org/wiki/?title=Rational_function&oldid=1297099759">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.

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