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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Conformal mapping in complex analysis</div> <p>In <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, a <b>Schwarz–Christoffel mapping</b> is a <a href="/wiki/Conformal_map" title="Conformal map">conformal map</a> of the <a href="/wiki/Upper_half-plane" title="Upper half-plane">upper half-plane</a> or the complex <a href="/wiki/Unit_disk" title="Unit disk">unit disk</a> onto the interior of a <a href="/wiki/Simple_polygon" title="Simple polygon">simple polygon</a>. Such a map is <a href="/wiki/Existence_theorem" title="Existence theorem">guaranteed to exist</a> by the <a href="/wiki/Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a> (stated by <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> in 1851); the Schwarz–Christoffel formula provides an explicit construction. They were introduced independently by <a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Christoffel</a> in 1867 and <a href="/wiki/Hermann_Amandus_Schwarz" class="mw-redirect" title="Hermann Amandus Schwarz"> Hermann Schwarz</a> in 1869. </p><p>Schwarz–Christoffel mappings are used in <a href="/wiki/Potential_theory" title="Potential theory">potential theory</a> and some of its applications, including <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surfaces</a>, <a href="/wiki/Poincare_disc_model#Artistic_realizations" class="mw-redirect" title="Poincare disc model">hyperbolic art</a>, and <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a polygon in the complex plane. The <a href="/wiki/Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a> implies that there is a <a href="/wiki/Biholomorphic" class="mw-redirect" title="Biholomorphic">biholomorphic</a> mapping <i>f</i> from the upper half-plane </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\zeta \in \mathbb {C} :\operatorname {Im} \zeta >0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>ζ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>:</mo> <mi>Im</mi> <mo>⁡</mo> <mi>ζ</mi> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\zeta \in \mathbb {C} :\operatorname {Im} \zeta >0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28075b7339fbed3492c4dc27538e15c91306c51c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.395ex; height:2.843ex;" alt="{\displaystyle \{\zeta \in \mathbb {C} :\operatorname {Im} \zeta >0\}}" /></span></dd></dl> <p>to the interior of the polygon. The function <i>f</i> maps the real axis to the edges of the polygon. If the polygon has interior <a href="/wiki/Angle" title="Angle">angles</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma ,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma ,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfa0423ad729cc9b43c1be6c65b94d86296aef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.907ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma ,\ldots }" /></span>, then this mapping is given by </p> <dl><dd><span class="mwe-math-element"><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(\zeta )=\int ^{\zeta }{\frac {K}{(w-a)^{1-(\alpha /\pi )}(w-b)^{1-(\beta /\pi )}(w-c)^{1-(\gamma /\pi )}\cdots }}\,\mathrm {d} w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo>−</mo> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⋯</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\zeta )=\int ^{\zeta }{\frac {K}{(w-a)^{1-(\alpha /\pi )}(w-b)^{1-(\beta /\pi )}(w-c)^{1-(\gamma /\pi )}\cdots }}\,\mathrm {d} w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb0c4943e30bbada3a5332e80af06f82882cb9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:59.62ex; height:6.843ex;" alt="{\displaystyle f(\zeta )=\int ^{\zeta }{\frac {K}{(w-a)^{1-(\alpha /\pi )}(w-b)^{1-(\beta /\pi )}(w-c)^{1-(\gamma /\pi )}\cdots }}\,\mathrm {d} w}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /></span> is a <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a>, and <span class="mwe-math-element"><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<b<c<\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo><</mo> <mi>c</mi> <mo><</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b<c<\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a11f2dc531f14ad0d2f32d395ae8cef22646f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.253ex; height:2.176ex;" alt="{\displaystyle a<b<c<\cdots }" /></span> are the values, along the real axis of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" /></span> plane, of points corresponding to the vertices of the polygon in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span> plane. A transformation of this form is called a <i>Schwarz–Christoffel mapping</i>. </p><p>The integral can be simplified by mapping the <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a> of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" /></span> plane to one of the vertices of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span> plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /></span>. Conventionally, the point at infinity would be mapped to the vertex with angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>. </p><p>In practice, to find a mapping to a specific polygon one needs to find the <span class="mwe-math-element"><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<b<c<\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo><</mo> <mi>c</mi> <mo><</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b<c<\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a11f2dc531f14ad0d2f32d395ae8cef22646f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.253ex; height:2.176ex;" alt="{\displaystyle a<b<c<\cdots }" /></span> values which generate the correct polygon side lengths. This requires solving a set of nonlinear equations, and in most cases can only be done <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerically</a>.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a semi-infinite strip in the <span class="texhtml"><var>z</var></span> <a href="/wiki/Complex_plane" title="Complex plane">plane</a>. This may be regarded as a limiting form of a <a href="/wiki/Triangle" title="Triangle">triangle</a> with vertices <span class="texhtml"><var>P</var> = 0</span>, <span class="texhtml"><var>Q</var> = π <i>i</i></span>, and <span class="texhtml"><var>R</var></span> (with <span class="texhtml"><var>R</var></span> real), as <span class="texhtml"><var>R</var></span> tends to infinity. Now <span class="texhtml">α = 0</span> and <span class="texhtml">β = γ = <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">π</span>⁄<span class="den">2</span></span></span> in the limit. Suppose we are looking for the mapping <span class="texhtml"><var>f</var></span> with <span class="texhtml"><var>f</var>(−1) = <var>Q</var></span>, <span class="texhtml"><var>f</var>(1) = <var>P</var></span>, and <span class="texhtml"><var>f</var>(∞) = <var>R</var></span>. Then <span class="texhtml"><var>f</var></span> is given by </p> <dl><dd><span class="mwe-math-element"><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(\zeta )=\int ^{\zeta }{\frac {K}{(w-1)^{1/2}(w+1)^{1/2}}}\,\mathrm {d} w.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>w</mi> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\zeta )=\int ^{\zeta }{\frac {K}{(w-1)^{1/2}(w+1)^{1/2}}}\,\mathrm {d} w.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b950972bcc0a2521be7ee78b84a73d97cd988f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:36.702ex; height:6.843ex;" alt="{\displaystyle f(\zeta )=\int ^{\zeta }{\frac {K}{(w-1)^{1/2}(w+1)^{1/2}}}\,\mathrm {d} w.\,}" /></span></dd></dl> <p>Evaluation of this integral yields </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(\zeta )=C+K\operatorname {arcosh} \zeta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mo>+</mo> <mi>K</mi> <mi>arcosh</mi> <mo>⁡</mo> <mi>ζ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(\zeta )=C+K\operatorname {arcosh} \zeta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5480a6b72968e6d4925c195f050cce643eda334b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.135ex; height:2.843ex;" alt="{\displaystyle z=f(\zeta )=C+K\operatorname {arcosh} \zeta ,}" /></span></dd></dl> <p>where <span class="texhtml"><var>C</var></span> is a (complex) constant of integration. Requiring that <span class="texhtml"><var>f</var>(−1) = <var>Q</var></span> and <span class="texhtml"><var>f</var>(1) = <var>P</var></span> gives <span class="texhtml"><var>C</var> = 0</span> and <span class="texhtml"><var>K</var> = 1</span>. Hence the Schwarz–Christoffel mapping is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\operatorname {arcosh} \zeta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>arcosh</mi> <mo>⁡</mo> <mi>ζ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=\operatorname {arcosh} \zeta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a18a6abef08c7811042504558d833dd5a8b170ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.793ex; height:2.509ex;" alt="{\displaystyle z=\operatorname {arcosh} \zeta .}" /></span></dd></dl> <p>This transformation is sketched below. </p> <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Schwarz-Christoffel_transformation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c2/Schwarz-Christoffel_transformation.png" decoding="async" width="580" height="177" class="mw-file-element" data-file-width="580" data-file-height="177" /></a><figcaption>Schwarz–Christoffel mapping of the upper half-plane to the semi-infinite strip</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Other_simple_mappings">Other simple mappings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=3" title="Edit section: Other simple mappings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Triangle">Triangle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=4" title="Edit section: Triangle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A mapping to a plane <a href="/wiki/Triangle" title="Triangle">triangle</a> with interior angles <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi a,\,\pi b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>π</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi a,\,\pi b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3954565a074874293a17b2a2edf67436db1553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.312ex; height:2.509ex;" alt="{\displaystyle \pi a,\,\pi b}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (1-a-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (1-a-b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8153fea30b8af770f052782e21af34f9f47afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.212ex; height:2.843ex;" alt="{\displaystyle \pi (1-a-b)}" /></span> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {dw}{(w-1)^{1-a}(w+1)^{1-b}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>w</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>a</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>w</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>b</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {dw}{(w-1)^{1-a}(w+1)^{1-b}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62531a53215531190c3abe69ff17ebd16a4faee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.002ex; height:6.676ex;" alt="{\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {dw}{(w-1)^{1-a}(w+1)^{1-b}}},}" /></span></dd></dl> <p>which can be expressed in terms of <a href="/wiki/Hypergeometric_function" title="Hypergeometric function">hypergeometric functions</a> or incomplete <a href="/wiki/Beta_function" title="Beta function">beta functions</a>. </p><p>The upper half-plane is mapped to a triangle with circular arcs for edges by the <a href="/wiki/Schwarz_triangle_map" class="mw-redirect" title="Schwarz triangle map">Schwarz triangle map</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Square">Square</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=5" title="Edit section: Square"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The upper half-plane is mapped to the square by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {\mathrm {d} w}{\sqrt {w(1-w^{2})}}}={\sqrt {2}}\,F\left({\sqrt {\zeta +1}};{\sqrt {2}}/2\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>w</mi> </mrow> <msqrt> <mi>w</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ζ</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {\mathrm {d} w}{\sqrt {w(1-w^{2})}}}={\sqrt {2}}\,F\left({\sqrt {\zeta +1}};{\sqrt {2}}/2\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/886dada57aaa671cedc89cf883128b70522a962e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.435ex; height:7.176ex;" alt="{\displaystyle z=f(\zeta )=\int ^{\zeta }{\frac {\mathrm {d} w}{\sqrt {w(1-w^{2})}}}={\sqrt {2}}\,F\left({\sqrt {\zeta +1}};{\sqrt {2}}/2\right),}" /></span></dd></dl> <p>where <i>F</i> is the incomplete <a href="/wiki/Elliptic_integral" title="Elliptic integral">elliptic integral</a> of the first kind. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Schwarzian_derivative" title="Schwarzian derivative">Schwarzian derivative</a> appears in the theory of Schwarz–Christoffel mappings.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.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("//upload.wikimedia.org/wikipedia/commons/6/65/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("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/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}}</style><cite id="CITEREFDriscoll" class="citation web cs1">Driscoll, Toby. <a rel="nofollow" class="external text" href="http://www.math.udel.edu/~driscoll/research/conformal.html">"Schwarz-Christoffel mapping"</a>. <i>www.math.udel.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-05-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.math.udel.edu&rft.atitle=Schwarz-Christoffel+mapping&rft.aulast=Driscoll&rft.aufirst=Toby&rft_id=http%3A%2F%2Fwww.math.udel.edu%2F~driscoll%2Fresearch%2Fconformal.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarz%E2%80%93Christoffel+mapping" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChristoffel1867" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Christoffel, Elwin Bruno</a> (1867). <a rel="nofollow" class="external text" href="https://zenodo.org/record/2358602">"Sul problema delle temperature stazionarie e la rappresentazione di una data superficie"</a> [On the problem of stationary temperatures and the representation of a given surface]. <i>Annali di Matematica Pura ed Applicata</i> (in Italian). <b>1</b>: <span class="nowrap">89–</span>103. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02419161">10.1007/BF02419161</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121089696">121089696</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annali+di+Matematica+Pura+ed+Applicata&rft.atitle=Sul+problema+delle+temperature+stazionarie+e+la+rappresentazione+di+una+data+superficie&rft.volume=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E89-%3C%2Fspan%3E103&rft.date=1867&rft_id=info%3Adoi%2F10.1007%2FBF02419161&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121089696%23id-name%3DS2CID&rft.aulast=Christoffel&rft.aufirst=Elwin+Bruno&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F2358602&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarz%E2%80%93Christoffel+mapping" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDriscollTrefethen2002" class="citation book cs1">Driscoll, Tobin A.; <a href="/wiki/Lloyd_N._Trefethen" class="mw-redirect" title="Lloyd N. Trefethen">Trefethen, Lloyd N.</a> (2002). <i>Schwarz–Christoffel Mapping</i>. Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511546808">10.1017/CBO9780511546808</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521807265" title="Special:BookSources/9780521807265"><bdi>9780521807265</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schwarz%E2%80%93Christoffel+Mapping&rft.pub=Cambridge+University+Press&rft.date=2002&rft_id=info%3Adoi%2F10.1017%2FCBO9780511546808&rft.isbn=9780521807265&rft.aulast=Driscoll&rft.aufirst=Tobin+A.&rft.au=Trefethen%2C+Lloyd+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarz%E2%80%93Christoffel+mapping" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchwarz1869" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Hermann_Schwarz" title="Hermann Schwarz">Schwarz, Hermann Amandus</a> (1869). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_journal-fuer-die-reine-und-angewandte-mathematik_1869_70/page/105">"Ueber einige Abbildungsaufgaben"</a> [About some mapping problems]. <i>Crelle's Journal</i> (in German). <b>1869</b> (70): <span class="nowrap">105–</span>120. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1869.70.105">10.1515/crll.1869.70.105</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121291546">121291546</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Crelle%27s+Journal&rft.atitle=Ueber+einige+Abbildungsaufgaben&rft.volume=1869&rft.issue=70&rft.pages=%3Cspan+class%3D%22nowrap%22%3E105-%3C%2Fspan%3E120&rft.date=1869&rft_id=info%3Adoi%2F10.1515%2Fcrll.1869.70.105&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121291546%23id-name%3DS2CID&rft.aulast=Schwarz&rft.aufirst=Hermann+Amandus&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_journal-fuer-die-reine-und-angewandte-mathematik_1869_70%2Fpage%2F105&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarz%E2%80%93Christoffel+mapping" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFForsyth1918" class="citation book cs1"><a href="/wiki/Andrew_Forsyth" title="Andrew Forsyth">Forsyth, Andrew Russell</a> (1918) [1st ed. 1893]. <i>Theory of Functions of a Complex Variable</i>. Cambridge.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Functions+of+a+Complex+Variable&rft.pub=Cambridge&rft.date=1918&rft.aulast=Forsyth&rft.aufirst=Andrew+Russell&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarz%E2%80%93Christoffel+mapping" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://archive.org/details/theoryoffunction028777mbp/page/n698/">§§267–270, pp. 665–677</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNehari1982" class="citation cs2">Nehari, Zeev (1982) [1952], <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/conformalmapping00neha"><i>Conformal mapping</i></a></span>, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61137-2" title="Special:BookSources/978-0-486-61137-2"><bdi>978-0-486-61137-2</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0045823">0045823</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conformal+mapping&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1982&rft.isbn=978-0-486-61137-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0045823%23id-name%3DMR&rft.aulast=Nehari&rft.aufirst=Zeev&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fconformalmapping00neha&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarz%E2%80%93Christoffel+mapping" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://archive.bridgesmathart.org/2016/bridges2016-179.pdf">The Conformal Hyperbolic Square and Its Ilk</a> Chamberlain Fong, Bridges Finland Conference Proceedings, 2016</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An analogue of SC mapping that works also for multiply-connected is presented in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCase2008" class="citation cs2">Case, James (2008), <a rel="nofollow" class="external text" href="https://archive.siam.org/pdf/news/1297.pdf">"Breakthrough in Conformal Mapping"</a> <span class="cs1-format">(PDF)</span>, <i>SIAM News</i>, <b>41</b> (1)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+News&rft.atitle=Breakthrough+in+Conformal+Mapping&rft.volume=41&rft.issue=1&rft.date=2008&rft.aulast=Case&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.siam.org%2Fpdf%2Fnews%2F1297.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarz%E2%80%93Christoffel+mapping" class="Z3988"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwarz%E2%80%93Christoffel_mapping&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/SchwarzChristoffelTransformation">"Schwarz–Christoffel transformation"</a>. <i><a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PlanetMath&rft.atitle=Schwarz%E2%80%93Christoffel+transformation&rft_id=http%3A%2F%2Fplanetmath.org%2FSchwarzChristoffelTransformation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchwarz%E2%80%93Christoffel+mapping" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://tobydriscoll.net/project/sc-toolbox/">Schwarz–Christoffel toolbox</a> (software for <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>)</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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