21 Conformal mapping II: the Schwarz–Christoffel mapping

21 Conformal mapping II: the 

Schwarz–Christoffel mapping 

Introduction 

In Chapter 16 we explored some of the geometrical properties of holomorphic functions, 

and in particular looked at the behaviour of the Möbius transformation. The key geometrical 

feature was that the mapping is conformal (where the derivative is non-zero) in the 

sense that it is locally angle-preserving. In Chapter 19 we highlighted the importance of 

conformal mapping to the solution of Laplace's equation in two dimensions. We produced 

several types of solution to Laplace's equation, including several examples where 

the region of interest was bounded by a circle or a line in the complex plane. 

A question that arise naturally is how to manage matters when the region is not a 

half-plane or interior/exterior of a circle. Here we must draw a sharp distinction between 

issues of general principle and practicalities of implementation. We shall begin by 

stating without proof an important, but non-constructive, theorem that addresses the 

general principle. Then we shall introduce the Schwarz–Christoffel (SC) mapping that 

gives an explicit construction for polygonal regions. 

There are very few examples of the SC mapping that can be worked out in 

closed-form in terms of ‘simple’ functions. A novel use of Mathematica is to use its 

advanced built-in special-function capabilities, and their linkage to the symbolic integrator, 

to give evaluations of several expressions usually left as complicated integrals in 

more traditional treatments. We can use such evaluations to facilitate the visualization of 

the mappings, and hence to confirm the correctness of the answer. As such this chapter 

contains an informal introduction to hypergeometric functions and elliptic functions, 

which arise naturally from SC mappings associated with triangles and rectangles. 

Mathematica's symbolic integrator also allows the results for triangles to be extended to 

regular n-gons, again in terms of hypergeometric functions. In more recent versions of 

Mathematica the implementation of hypergeometric functions of two variables permits a 

still larger category of mappings to be explored. It should be emphasized that this is a 

purely analytical treatment, best used as a teaching tool. 

For full professional use there is an extensive and highly developed numerical 

treatment already available. We shall also comment briefly at the numerical implementation 

of the SC mapping. Gratitude is expressed to Professor L. N. Trefethen F.R.S. for 

supplying background material on this topic. Readers interested in pursing SC mapping 

in more detail, whether analytical or numerical, are encouraged to see Section 2.17 and 

to consult the definitive and up-to-date text on the matter: Driscoll and Trefethen (2002). 

This chapter is best regarded as an educational bridge between the older and very limited 

analytical discussions in textbooks and the full numerical approach developed by 

Trefethen, Driscoll and co-workers. Our bridge is constructed using the advanced 

analytical tools in Mathematica.

4H2 =o,-+$8&;n)+>(i(&2it#&?)t#$,)tic) 

21.1 The Riemann mapping theorem 

This theorem is about simply connected domains ! in ! that are not the whole of !. 

7iven such a domain the theorem states that t#$r$ &i( &) 1-1 #o+o,or-#ic &/unction 

2 = / !3L t#)t&,)-( ! onto&t#$&int$rior&o/&t#$&unit&di(56&i7$7&to&t#$&($t&o/&co,-+$8&nu,9$r( 

2 (uc#&t#)t # 2 # " 1. If we further specify that a given point 3 0 satisfies 0 = / !3 0L, and 

that a specified direction at 30 is mapped into a specified direction at 0 (for example the 

direction along the positive real axis), the mapping is unique. 

The proof of this result is rather beyond the scope of this book. A reasonably 

accessible treatment is given by Eettman (196H). Iote that the result could just as easily 

have been stated about the existence of a mapping into the upper half-plane, by composing 

/ with an appropriate Möbius transform. 

The drawback of this theorem is that it is purely about existence M it tells us 

nothing about how to explicitly construct the mapping. 

21.2 The Schwarz–Christoffel transformation 

w 1 

" 

! 1 ! 2 

! H 

! 3 

w 3 

w 2 

w H 

! 4 

w 4 

w 

# 

$ 

% 1 % 2 % 3 % 4 % H 

This is a very explicit constructive method for finding mappings that will cope with 

regions with polygonal boundaries. We can work with various base regions, such as the 

upper half-plane or interior of a circle, and to state the basic results we shall consider a 

mapping from the upper half-plane. 

Suppose that we have a polygon in the 2-plane with vertices at 21, 22, S, 2n 

and interior angles at those vertices # 1, # 2, S, # n. Suppose that these points map onto 

points 81, 82, S, 8n on the real axis of the 3-plane. Let ! denote the interior of the 

polygon and : the interior of the upper half-plane. Then a transformation that maps : 

onto ! is given by the SchwarzMChristoffel (SC) formulaW 

2 = ; $ !3 $ 81L !#1%%L$1 !3 $ 82L !#2%%L$1 &S!3 $ 8nL !#n %%L$1 &' 3 + < 

In differential form this is just 

(21.1)

45.63%73*8+/.8+22'%,.99:.)-0.; so that ! moEes in a ()*+',-)./'%0$ 'his is why the ima:e in the ! 

./ane is .o/y:ona/$ Now we need to see what ha..ens at the corners$ Ghen " .asses 

throu:h $'> a// the terms e=ce.t those inEo/Ein: $' remain constant> Fut the factor 

Ar:!" # $' " 

Ium.s from a Ea/ue of %> when " ( $ ' > to a Ea/ue of J> when $ ) $ ' $ 'he term 

$ $' 

!!!!!!! # #% Ar:!" # $'" 

% 

!2#$H% 

!2#$K% 

therefore Ium.s from the Ea/ue $' # % to the Ea/ue J$ 'he direction in which ! is traEe/L 

/in: therefore rotates in a .ositiEe sense !i$e$ antiLc/oc;wise%> Fy % # $'$ A chan:e in 

direction Fy this amount corres.onds to the introduction of a corner in the .o/y:on with 

interior an:/e $'$ 

Ghat this discussion estaF/ishes is that the rea/ a=is ma.s to the Foundary of a 

.o/y:ona/ re:ion$ 'his wi// Fe sufficient for our .ur.oses$ Mne can :o further and 

estaF/ish that the ma..in: is oneLtoLone and indeed ta;es the 1220* ha/fL./ane to the 

'%)0*'3* of a .o/y:on$ Ge can a/ways chec; this /atter .oint Fy wor;in: out sam./e 

Ea/ues of .articu/ar ma..in:s> such as the ima:e of *$ 

! 4he point at in#inity7 a simpli#ication 

Ge can assume that one of the .oints on the rea/ a=is> say $ % is at infinity> with it then 

Fein: dro..ed from the /ist of .oints in the transformation$ 'o see this> we write> with $% 

finite> 

# 

# " 

+ 

!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!! 

!#$%" !$%#%"## 

!! 

!!!!!!!! 

!" " #+ !" # $ #" !$# #%"## !" # $ 2" !$2 #%"## &$ $% # " 

!!!!!!!!!!!!!!!! 

$% 

% !$ %#%"## 

!2#$N% 

!2#$O%

TUT ./*0+1!2#",+3()(2$)4526,451*,4)7, 

!"#$#%$&'(%$&)%$*+,+&$!" ! "-$(%%.+/0$#1$f+/+&%$'/3$&)%$*'4&$f'5&"6$36".4$"7&-$*%'8+/0$74 

#+&) 

!$ 

######## 

!% $ #% !% & ! ;L !f ; #(L&; !% & !:L !f : #(L&; ?!% & !"&;L !f "&;#(L&; 

! The exponents and other comments 

9:; 

@%$5'/$,'(%$'$4+,.*%$0%",%&6+5'*$'607,%/&$&)'&$.7&4$"/%$5"/4&6'+/&$$"/$&)%$%A."/%/&4< 

!"&%$&)'&$&)%$47,$"f$&)%$%A&%6+"6$'/0*%4$"f$'/B$5*"4%3$."*B0"/$+4$:(

)*+&,-.,rm!l+m!223-g+556+789+:;8"!r$

MNO Complex Analysis with Mathematica 

!a#te&ia(!)(*)#ma,!*-(c/0 

1#a(2e/0 3#a(2e/0 )4ti)(&///" 5! 

6h)8!9#a4hic&:##a3!# 

!a#te&ia(Ma4!< =0 1#a(2e0 3#a(2e0 

)4ti)(&0 >i&4,a3?-(cti)( "# @Ae(tit3"0 

!a#te&ia(Ma4!*-(c0 1#a(2e0 3#a(2e0 

)4ti)(&0 >i&4,a3?-(cti)( "# @Ae(tit3" 

$"0 >i&4,a3?-(cti)( "# B>i&4,a3?-(cti)("C 

!"#$%&'(()&*#&#+,%&-,.%#&"/*01'"&2,#+&a ! 34&5"6*''&#+*#&!a#te&ia(!)(*)#ma,&%+(2% 

#+"&,0*7"%&(-&#+"&."*'&*89&,0*7,8*.:&6((.9,8*#"&',8"%&;89".&#+"&0*11,874& 

!a#te&ia(!)(*)#ma,!%D & Ei :#c6i(!;%#&#2(&?".#,6"%&#( 

@"&6(8%,9"."94&A8&,01(.#*8#&6*%"=&'"-#&-(.&:(;&#(&,8?"%#,7*#"&,8&9"#*,'&,8&B/".6,%"&C343=&,% 

2+"."&#+"&1(':7(8&+*%&#2(&.,7+#-*87'"9&#;.8%&(-&(11(%,#"&%"8%"&E&#+,%&'"*9%&#(&*&0*11,87 

-.(0&#+"&;11".&+*'--1'*8"&#(&(8"&2,#+&*&%#"1-%+*1"9&@(;89*.:4 

2)*+ !riangular and rectangular boundaries 

21 Conformal mapping II: the Schwarz–Christoffel mapping 457 

Show@Graphics @88GrayLevel @0.6D, Polygon@880, 0

458 Complex Analysis with Mathematica 

Integrate@z ^Ha ê Pi - 1L H1 - zL ^ Hb ê Pi - 1L, 

8z, 0, 1 0, b > 0


Now we ask about this function. 

? Beta 

Beta@a, bD gives the Euler beta function BHa, bL. 

Beta@z, a, bD gives the incomplete beta function BzHa, bL. 

Mathematica has now given us a result in terms of the incomplete beta function. The 

theory of analytic continuation then can be used to argue that this holds everywhere. A 

proper discussion of this is rather beyond the scope of this text, but, basically, if two 

holomorphic functions agree on a suitable region, they are equal everywhere, so we can 

use the formula in terms of a beta function, which is holomorphic, for other values of z 

besides those suggested by the output from the integrator. In any case, we can check this 

formula by using our visualization tool. We consider an equilateral triangle, with 

a = b = p 

ÅÅÅÅÅ 

3 

The formula for w is then: 

w ê. 8a -> Pi ê 3, b -> Pi ê 3< 

BzH 1 

ÅÅÅÅ , 3 1 

ÅÅÅÅ L GH 3 2 

ÅÅÅÅ L 3 

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ 

ÅÅÅÅ 

GH 1 

L 3 2 

We define a Mathematica formula by 

g@z_D = Simplify@%D; 

(21.23) 

Now let us take a look at the result (this needs a fast machine to compute reasonably 

quickly): 

CartesianConformal@Hg@#D &L, 8-8, 8 1000, 

PlotRange -> All, PlotStyle Ø AbsoluteThickness@0.01DD 

8 

6 

4 

2 

-7.5 -5 -2.5 2.5 5 7.5 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0.2 0.4 0.6 0.8 1 

Slight numerical plotting effects at end-points aside, this confirms the validity of the 

closed-form answer. 

Another interesting exercise is to consider the region in the upper half plane 

above a related triangle and the real axis – this is left for you to explore in Exercise 21.2.


‡ Ÿ Hypergeometric and beta functions 

The incomplete beta function is a special case of the hypergeometric function. You can 

just ask the kernel for some basic information about such functions 

? Hypergeometric2F1 

Hypergeometric2F1@a, b, c, zD 

is the hypergeometric function 2F1Ha, b; c; zL. 

More useful is the series expansion, which extrapolates the pattern shown here for the 

first few terms: 

Series@ Hypergeometric2F1@a, b, c, zD, 8z, 0, 3 1 in the following 

example. In this case the differential SC mapping takes the form


!w 

ÅÅÅÅÅÅÅÅ 

!z = A Hz + aL-1ê2Hz + 1L -1ê2 Hz - 1L -1ê2 Hz - aL -1ê2 

A 

= - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 

è!!!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 

1 - z2 è!!!!!!!!!!!!!! 

a2 - z2 (21.24) 

It is a little more convient to write this in terms of a = 1 ê a and a new overall constant 

A £ as 

!w 

ÅÅÅÅÅÅÅÅ 

!z = 

A £ 

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 

è!!!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 

1 - z2 è!!!!!!!!!!!!!!!!! 

1 - a2 z2 (21.25) 

The integral of this is only moderately exotic, in that it defines one of the well known 

family of elliptic functions. We now set A £ = 1 and evaluate the integral with some 

suitable Assumptions: 

Integrate@1 ê HSqrt@1 - z ^2D * Sqrt@1 - a ^2 z ^2DL, 

8z, 0, z 0


CartesianConformal@HSCRect@#, 1 ê 2D &L, 

8-16, 16 200, PlotRange -> AllD 

15 

12.5 

10 

7.5 

5 

2.5 

-15 -10 -5 5 10 15 

2 

1.5 

1 

0.5 

-1.5-1-0.5 0.5 1 1.5 

We can see very clearly that the EllipticF function is effecting the mapping we 

want. The four vertices are a the points !b, !b + Â c, where, as functions of the parameter 

a = 1 ê a, b is given by integrating the mapping from 0 to 1: 

b @a_D = SCRect@1, aD 

KHa 2 L 

where the Mathematica function just output is given by 

InputForm@%D 

EllipticK[a^2] 

For the case we have plotted, the value of this is: 

b@1 ê 2D êê N 

1.68575 

To extract the right value of c we need to be slightly careful with the branching properties 

(also see Wolfram (1993)) of the EllipticF function – to this end we define: 

c@a_, e_D = SCRect@1 ê a + e I, aD - b@aD 

FJsin -1 JÂ e + 1 

ÅÅÅÅÅ 

a N À a 2 N - KHa 2 L 

For the example at hand, the most obvious approach does not quite work: 

c@1 ê 2, 0D êê N 

-4.00238 µ 10 -9 - 2.15652 Â 

If we nudge the point slightly into the upper half-plane, which is, after all, where we are 

working, we get:

21 Conformal mapping II: the Schwarz–Christoffel mapping

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