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| In [[complex analysis]], a '''Schwarz–Christoffel mapping''' is a [[conformal transformation]] of the [[upper half-plane]] onto the interior of a simple [[polygon]]. Schwarz–Christoffel mappings are used in [[potential theory]] and some of its applications, including [[minimal surface]]s and [[fluid dynamics]]. They are named after [[Elwin Bruno Christoffel]] and [[Hermann Amandus Schwarz]].
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| ==Definition==
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| Consider a polygon in the complex plane. The [[Riemann mapping theorem]] implies that there is a [[bijective]] [[biholomorphic]] mapping ''f'' from the upper half-plane
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| :<math> \{ \zeta \in \mathbb{C}: \operatorname{Im}\,\zeta > 0 \} </math>
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| to the interior of the polygon. The function ''f'' maps the real axis to the edges of the polygon. If the polygon has interior [[angle]]s <math>\alpha,\beta,\gamma, \ldots</math>, then this mapping is given by
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| :<math>
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| f(\zeta) = \int^\zeta \frac{K}{(w-a)^{1-(\alpha/\pi)}(w-b)^{1-(\beta/\pi)}(w-c)^{1-(\gamma/\pi)} \cdots} \,\mbox{d}w
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| </math>
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| where <math>K</math> is a [[Constant (mathematics)|constant]], and <math>a < b < c < ...</math> are the values, along the real axis of the <math>\zeta</math> plane, of points corresponding to the vertices of the polygon in the <math>z</math> plane. A transformation of this form is called a ''Schwarz–Christoffel mapping''.
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| It is often convenient to consider the case in which the [[point at infinity]] of the <math>\zeta</math> plane maps to one of the vertices of the <math>z</math> plane polygon (conventionally the vertex with angle <math>\alpha</math>). If this is done, the first factor in the formula is effectively a constant and may be regarded as being absorbed into the constant <math>K</math>.
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| ==Example==
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| Consider a semi-infinite strip in the {{math|<VAR >z</VAR >}} [[complex plane|plane]]. This may be regarded as a limiting form of a [[triangle]] with vertices {{math|<VAR >P</VAR > {{=}} 0}}, {{math|<VAR >Q</VAR > {{=}} π ''i''}}, and {{math|<VAR >R</VAR >}} (with {{math|<VAR >R</VAR >}} real), as {{math|<VAR >R</VAR >}} tends to infinity. Now {{math|α {{=}} 0}} and {{math|β {{=}} γ {{=}} {{frac|π|2}}}} in the limit. Suppose we are looking for the mapping {{math|<VAR >f</VAR >}} with {{math|<VAR >f</VAR >(−1) {{=}} <VAR >Q</VAR >}}, {{math|<VAR >f</VAR >(1) {{=}} <VAR >P</VAR >}}, and {{math|<VAR >f</VAR >(∞) {{=}} <VAR >R</VAR >}}. Then {{math|<VAR >f</VAR >}} is given by
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| :<math> f(\zeta) = \int^\zeta | |
| \frac{K}{(w-1)^{1/2}(w+1)^{1/2}} \,\mbox{d}w. \, </math>
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| Evaluation of this integral yields
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| <!-- :<math> z = f(\zeta) = C + K \operatorname{arccosh}\,\zeta, </math>-->
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| :{{math|''z'' {{=}} ''f''(''ζ'') {{=}} ''C'' + ''K'' arccosh ''ζ''}}
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| where {{math|<VAR >C</VAR >}} is a (complex) constant of integration. Requiring that {{math|<VAR >f</VAR >(−1) {{=}} <VAR >Q</VAR >}} and {{math|<VAR >f</VAR >(1) {{=}} <VAR >P</VAR >}} gives {{math|<VAR >C</VAR > {{=}} 0}} and {{math|<VAR >K</VAR > {{=}} 1}}. Hence the Schwarz–Christoffel mapping is given by
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| <!--:<math> z = \operatorname{arccosh}\,\zeta. </math>-->
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| :{{math|''z'' {{=}} arccosh ''ζ''}}
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| This transformation is sketched below.
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| [[Image:Schwarz-Christoffel transformation.png|frame|center|Schwarz–Christoffel mapping of the upper half-plane to the semi-infinite strip]]
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| ==Other simple mappings==
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| ===Triangle===
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| A mapping to a plane [[triangle]] with angles <math>\pi a,\, \pi b</math> and <math>\pi(1-a-b)</math> is given by
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| :<math>z=f(\zeta)=\int^\zeta \frac{dw}{(w-1)^{1-a} (w+1)^{1-b}}.</math>
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| ===Square===
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| The upper half-plane is mapped to the square by
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| :<math>z=f(\zeta) = \int^\zeta \frac {\mbox{d}w}{\sqrt{w(w^2-1)}}
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| =\sqrt{2} \, F\left(\sqrt{\zeta+1};\sqrt{2}/2\right),
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| </math>
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| where F is the incomplete [[elliptic integral]] of the first kind.
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| ===General triangle===
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| The upper half-plane is mapped to a triangle with circular arcs for edges by the [[Schwarz triangle map]].
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| ==See also==
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| * The [[Schwarzian derivative]] appears in the theory of Schwarz–Christoffel mappings.
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| ==References==
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| *{{Citation | last1=Driscoll | first1=Tobin A. | last2=Trefethen | first2=Lloyd N. | author2-link=Lloyd N. Trefethen | title=Schwarz-Christoffel mapping | url=http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521807265 | publisher=Cambridge University Press | series=Cambridge Monographs on Applied and Computational Mathematics | isbn=978-0-521-80726-5 | id={{MathSciNet | id = 1908657}} | year=2002 | volume=8}}
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| *{{Citation | last1=Nehari | first1=Zeev | title=Conformal mapping | origyear=1952 | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-61137-2 | id={{MathSciNet | id = 0045823}} | year=1982}}
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| ==Further reading==
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| * {{Citation | last1=Case | first1=James | title=Breakthrough in Conformal Mapping | url=http://siam.org/pdf/news/1297.pdf | year=2008 | journal=SIAM News | volume=41 | issue=1}}.
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| ==External links==
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| * {{planetmath reference|id=6289|title=Schwarz-Christoffel transformation}}
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| * [http://math.fullerton.edu/mathews/c2003/SchwarzChristoffelMod.html Schwarz–Christoffel Module by John H. Mathews]
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| * [http://www.math.udel.edu/~driscoll/SC Schwarz–Christoffel toolbox] (software for [[MATLAB]])
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| {{DEFAULTSORT:Schwarz-Christoffel mapping}}
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| [[Category:Conformal mapping]]
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