Potential and Field in a 
Plane Condensor with 
One Split Electrode

Long write-up

St. Rosegger, B. Schnizer, G. Schweitzer

Institut fuer Theoretische Physik
Technische Universitaet Graz
A-8010 Graz, Austria, Europe
Email: schnizer@itp.tu-graz.ac.at

2004-12-07

Please report errors and difficulties per email to the adress given above in the affiliation.

1.  Description of the notebooks

1.1  Description of the program in the notebook SplitCond.nb

The potential and field distributions for the configuration shown below, a twodimensional infinite plane condensor 
of width  D, whose upper electrode has a gap of width  g,  are computed by conformal maps. The upper left electrode 
is at  Voltage   scV ;  the other electrodes are grounded. The potential and the field are plotted, as is the field on the
electrodes. The values of the potential and field are computed at a point which may be selected by the user. The relative
current  I/qv  induced on the upper left electrode by a charge q moving with constant velocity  v  on various trajectories 
parallel to the y-axis is shown as function of y.  The user may also choose the x-coordinate of such a trajectory.

The evaluations just described are done with the help of a conformal map, W =  f(z), mapping the configuration above 
to a common plane condensor with a homogeneous field. In more detail: By this map  the  upper half of the z = x + i y -
plane comprising the above configuration is mapped to a strip of width   π   in the  W = U + i V -plane; whose upper 
boundary,  V = π, represents an electrode at potential  Φ= scV, while the lower one,  V = 0,  is grounded, Φ = 0.

In this notebook the strict naming of variables prescribed in the notebook  SplitCond.nb is not observed. So it is not
recommended to run it in the kernel at the same time as an other notebook, even not together with  SplitCond.nb.

The potential difference between the electrodes is denoted as (Subscript[V, 0]).  Particular values are assigned to most variables
for drawing pictures.  If these are changed by the user, the resulting figures may be unsatisfactory.

1.2  Description and content of the notebook SplitCondLW.nb

By the conformal map: 
				W = f(z),  	z =  (f^(-1))(W)								(1)

described briefly in the preceeding paragraph  the  upper half of the z = x + i y  -plane comprising the above configuration 
is mapped to a closed infinite plane condensor of width   π   in the   W = U + i V -plane; whose upper electrode  V = π   is at 
potential  scV, while the lower one,  V = 0,  is grounded.   
This map is a sequence of three maps, which are detailedn in the subsubsections below.

1.2.1 The map   z = (Subscript[f, 3])(t').

The first of the  three maps is a Schwarz -Christoffel transformation:

					z = (Subscript[f, 3])(t')  =  (D/π)[ ln t' + ((((1 - p))^2)/p)((1/2)- (1/(1+t')) ) ] ,				(2)

which maps the half space y  ≥  0, so the configuration under consideration,  to the half space s' ≥ 0 of the 
complex  t' = r' + i s' -plane such that the lines  y = 0  and  y = D  become the axis  s' = 0.  The parameter  p  depends 
on  the widths  D  of the condensor and on the width  g of the gap; more precisely it depends on the ratio  g/D only,  
since it is the solution of the following trancendental equation:

						ln p + ((  (p^2)  -  1)/(2 p))= ((π g)/(2 D))							(3)

This solution is found by a numerical method, e.g. by Newton's method. The starting value is uncritical. The mapping 
function  (Subscript[f, 3])(t') depends on  p only  via the parameter 

						α  =  ((((1  -   p))^2)/p).							(4)
												
Inversion of the above formula leads to a two-valued function.  The branch needed in the present application is given 
by (cf. below subsect. 3.3.4):
					p  =  (1/2) (2 + α + (Sqrt[α]) (Sqrt[4 + α])).						(5)
					
No exact analytical expression can be found for the inverse t' = (Subscript[f, 3]^(-1))(z) of the function z = (Subscript[f, 3])(t').  So this inverse function
must be found by a numerical method, e.g. Newont's method. A suitable starting value must be chosen, which depends 
on the value of z. These starting values zs  were found by trial and error using the insight gained from the figures at the 
end of subsect. 3.4.3; they are:
					
				Re(z)  ≥ 0  ∧  Im(z)  ≥  D:  zs =  - (α + 1)/2 + 0.2 i ;
				Re(z)  < 0  ∧  Im(z)  ≥  D:  zs =  - (1/α + 1)/2 + 0.2 i;
				Re(z)  ≤ 0  ∧  Im(z)  <  D:  zs =     0.2 i ;						(6)
				Re(z)  > 0  ∧  Im(z)  <  D:  zs =     1.2 i .

For some domains the convergence of this procedure is not assured. In particular,
for  Re(z) < - g/2 and  0 ≤ Im(z) ≤ D,  it may be difficult to find a good starting value; in addition, default accuracy may 
even fail in FindrRoot[]. An approximation for this inverse valid for sufficiently negative Re(z) is derived below in 3.5
by iteration.

1.2  .2 The map   t'= (Subscript[f, 2])(t).

This map is a Möbius transformation, the ratio of two linear functions:

			 t' = (Subscript[f, 2])(t)=(t/(1 - t)),    t  =  (Subscript[f, 2]^(-1)) (t') = ((t')/(1+   t')).						(7)

 maps the t- on the t'-plane such that the s- becomes the s'-axis.

1.2.3 The map   t = (Subscript[f, 1])(W).

The function:
			   W = (Subscript[f, 1]^(-1))(t)  =  ln t, 	t = (Subscript[f, 1]^(-1))(W) = e\(TraditionalForm\`\^W\) 							(8)
			   
maps the strip  0 ≤ Im(W) ≤ π, so a simple plane infinite condensor,  to the  t = r + i s -plane such that the upper 
electrode V = D becomes the ray  s = 0,  r <  0; the lower electrode V = 0 the ray    s = 0,  r  >  0.

2. Numeric Values for Variables

The main purpose of this long write-up is to describe the conformal map(s). For most purposes (drawings,
numeric values) the following numeric values are chosen for the parameters.

Needs["Graphics`Colors`"]

gd  =  
        1    ;                  
g  =  
        1.2 ;                   
ep  =  
        10. ^-9 ;        
ra  =  
        3  ;                   
V0  =  
        1 ;

3. The Third Map,  z =  (Subscript[f, 3])(t').

This is a Schwartz-Christoffel transformation, which maps the half plane  Im(t') = s' ≥ 0 to the half plane 
 Im(z) = y ≥ 0 such that the r'-axis goes over into the three electrodes at y = 0, -∞ ≤ x ≤ ∞, 
 at y = D, -∞ ≤ x ≤ -g/2 and at  y = D, g/2 ≤ x ≤ ∞. This is shown below in detail.

3.1  The Real Configuration

up  =  {Thickness[ .01], Hue[0], 
            Line[{{-ra, gd}, {-g/2, gd}}],  Hue[ .6], 
            Line[{{ra, gd}, {g/2, gd}}]} // Graphics ;

lo  =  {Thickness[ .01], Line[{{-ra, 0}, {ra, 0}}]} // 
          Graphics;

cou  = 
        Show[up, lo, Axes  ->  True,  
          AspectRatio  ->  Automatic, 
          PlotRange  ->  {0, 
              2.5}, 
AxesLabel  ->   {"x", "y\
"},  ImageSize  ->  550];

3.2 Graphics for Head

gg  =  2;

uph  =  {Thickness[ .01], Line[{{ra, gd}, {gg/2, gd}}], Hue[0], 
            Line[{{-ra, gd}, {-gg/2, gd}}]} // 
          Graphics ;
schr  =  {Text["<  g/2  >", { .25 gg, 
              gd}], 
Text["∧", {0.5 ra, 
              0.9 gd}], Text["∨", {0.5 ra, 0.1 gd}], 
          Text["D", {0.5 ra, 
              0.5 gd}], 
Text["Φ = 0", { 
              0.5 ra, 1.1 gd}], 
          Text["Φ = 0", { 0.2 ra, 0.1  gd}],  
          Hue[0], 
Text["Φ = scV", {- 
                  0.6 ra, 1.1 gd}]};

\(\(Show[uph, lo, schr // Graphics, Axes\  -> \ True, 
        Ticks\  -> \ 
          None, 
AspectRatio\  -> \  .4, 
        AxesLabel\  -> \ {"x", "y"}, \ 
        PlotRange\  -> \ {\(-0.1\), \ 
            1.3\ gd}\ , \ 
TextStyle\  -> \ {FontSize\  -> \ 12}, \ ImageSize\  -> \ \ 520];\)\)

3.3 The Parameters p = f(g/D)  > 0  and  α = (((p  -  1))^2)/p

The parameter  p  is determined by the configuration of the electrodes as shown above. 
As a solution of the transcendental equation (3) it depends only on the ratio gg = g/D.
Various approximate formulae for the function  f(g/D)  could be derived and discussed. There 
is no real need for them since the value  p  belonging to a given ratio  g/D can be obtained
rather easily by a numeric method. The mapping function does not contain the parameter
p but the parameter  α = (((p  -  1))^2)/p. 
Graphs are given displaying the dependence of these two parameters on g/D.

3.3.1  The characteristic parameter  p.

Eq.(3) determines the characteristic parameter p:

Clear[gg]
fr[gg_] =  
      Log[p]  +  (p^2  -  1)/(2  p)  -  π gg/2

\(\(pp\  = \ 
        Plot[\ p\  /. \ FindRoot[fr[gg], {p, 1}], {gg, 0, 
            10}, 
AxesLabel\  -> \ {"g/D", "p"}];\)\)

3.3.2  The  parameter  α  contained in the mapping function.

Clear[α]
eq  =    α  == (((p  -  1))^2)/p;

sap  =  Solve[eq, p]

3.3.4  The  parameter  p as a function of  α  contained in the mapping function

pa  =  
        Plot[   (((p  -  1))^2)/p  /.  
            FindRoot[fr[gg], {p, 1}], {gg, 0, 
            10}, 
AxesLabel  ->  {"g/D", "\
α"},  PlotStyle  ->  Dashing[{ .01}]];

3.3.3  The  parameters  p  and  α versus the ratio  g/D

\(\(figpa\  = \ 
        Show[pp, pa, \ 
          AxesLabel\  -> \ {"g/D", "p, α"}, \ 
          ImageSize\  -> \ \ 550, 
Epilog\  -> \ {Text["p", {5, 13}], Text["α", {5, 7}]}, \ 
          TextStyle\  -> \ {FontSize\  -> \ 14}];\)\)

3.3.4  The  parameter  p as a function of  α  contained in the mapping function

\(\(Plot[Evaluate[p\  /. \ sap], {α, 0, 25}, 
        AxesLabel\  -> \ {"α", "p"}, 
ImageSize\  -> \ 550];\)\)

The graph above shows p  as a function of  α is given by 

					p  =  (1/2) (2 + α + (Sqrt[α]) (Sqrt[4 + α]))

3.4  Investigating the map for a particular value of g/D ( = 1.3)

The insight gained from this investigation will help us to find the approximation to the 
inverse map t' = \(TraditionalForm\`\(\(f\_3\%\(-1\)\)(
          z)\)\ \ where\ the\ common\ numeric\ \ procedure\ \(\(fails\)\(.\)\)\)

3.4.1  Numeric values for the map

g  =  1.3;
g/gd
fr[g];

3.4.2  The parameter p and the mapping function

vp  =  p  /.  FindRoot[fr[g], {p, 1}]

α  =  ((((p  -  1))^2)/p)  /.  p  ->  vp

fz[t_]  =  
gd/π  (Log[
                t]  +  α (1/2.   - 1/(1  +   t)) ) // 
        Expand

3.4.3 Various points and intervals in the t-plane and their images in the z-plane

The naming of the points follows the description of the Schwartz-Christoffel transformation
given by Wendt [3], pp. 74 and 76. These points and their images are shown in the graphics at the end
of this  section.

vzsu  =  gd 1.1 ;    
vzsl  =  
        gd 0.9 ;

point E

-1/vp

zzp  = fz[-1/vp  +  I ep ] // Chop

pte  =  Point[{-1/vp,  0}] ;
tte  :=  
        Text["E", {-1/vp, vtst}];
pze  =  Point[{Re[zzp], Im[zzp]}];
tze  =  Text["E", {Re[zzp], Im[zzp] vzsu}];

point B

-vp

zzp  =  fz[-vp   +  I ep] // Chop

ptb  =  Point[{-vp,  0}];
ttb  :=  
        Text["B", {-vp, vtst}];
pzb  =  Point[{Re[zzp], Im[zzp]}];
tzb  =  Text["B", {Re[zzp], Im[zzp] vzsu}];

a point in the range (B, E)

(-vp  - 1/vp)/2.

fz[(-vp  - 1/vp)/2.   +  I ep] // Chop

points in the range (A, B)

-vp - 1

fz[-vp  -  1  +  I ep] // Chop

-vp - 2

fz[-vp  -  2  +  I ep] // Chop

point D

fz[-1  +  I ep] // Chop

a point in the range (0, H)

fz[0.5]

point H

zzp  = fz[1]

pth  =  Point[{1,  0}];
tth  :=  Text["H", {1, vtst}];
pzh  =  Point[{Re[zzp], Im[zzp]}];
tzh  =  
        Text["H", {Re[zzp], 
            Im[zzp] + (vzsu  -  gd)}];

a point in the range (H,A')

fz[1.5]

the origin t' = 0 is mapped to z = -∞

fz[0]

a point slightly above t' = 0

fz[I ep]

3.4.3.1  Mapping the negative r-axis

fz[tt]

lrn  = 
        Table[{{Re[r], 
              Im[r]}, {fz[
                r] }}, {r, -3.05, -0.05,  .1}];

zrn  =  
        Partition[
          Map[{Re[#], Im[#]} &, Transpose[lrn][[2]], {2}] // Flatten, 
          2];

ListPlot[zrn,  PlotRange  ->  All, 
        PlotJoined  ->  True, PlotStyle  ->  Hue[0], 
        AxesLabel  ->  {"x", "y"}];

This graph shows that the negative  ray of the r-axis is mapped to the line z = i gd.
The same result is seen from the table below.

trn =  
        Partition[Transpose[lrn][[1]] // Flatten, 2];
Print[" r'", "      s'", "      x", "             \
y"]
TableForm[Map[Flatten, {trn,  zrn} // Transpose, {1}]]

3.4.3.2  Mapping the positive r-axis

fz[tt]

lrp  = 
        Table[{{Re[r], Im[r]}, {fz[r] }}, {r, 0.05, 
            3.05,  .1}];

zrp  =  
        Partition[
          Map[{Re[#], Im[#]} &, Transpose[lrp][[2]], {2}] // Flatten, 
          2];

ListPlot[zrp,  PlotRange  ->  All,  
        PlotJoined  ->  True, 
PlotStyle  ->  
          Yellow, AxesLabel  ->  {"x", "y"}];

This graph shows that the positive  ray of the r-axis is mapped to the line y = 0.
The same result is seen from the table below.

trp =  
        Partition[Transpose[lrp][[1]] // Flatten, 2];
Print[" r'", "      s'", "      x", "             \
y"]
TableForm[Map[Flatten, {trp,  zrp} // Transpose, {1}]]

3..4.3.3  The Gap, its image in the t'-plane

lg  =  
        Table[{zz  = - g/2  +  k  .025  +  gd I,  
              soz  =  
                t  /. FindRoot[fz[t]  - zz, {t, - .4  +  I  .1}, 
                    MaxIterations  ->  30], fz[soz] - zz}, {k, 0, 
              42}] // Chop ;

lk  =  Table[{zz  =  g/2  -  k  .005  +  gd I,  
                soz  =  
                  t  /. FindRoot[fz[t]  - zz, {t, -2  +  I  .1}, 
                      MaxIterations  ->  30], fz[soz] - zz}, {k, 0, 
                28}] // Chop // Reverse;

vp

lz  =  Transpose[Join[lg, lk]][[2]];

nl  =  {Transpose[Join[lg, lk] // Re][[1]], lz} // 
          Transpose;

fl  =  {Transpose[Join[lg, lk] // Re][[1]], lz} // 
          Transpose;

lc  =  {Re[Transpose[fl][[2]]], 
            Im[Transpose[fl][[2]]]} // Transpose;

\(\(pg\  = \ 
        ListPlot[lc, \ PlotJoined\  -> \ True, \ 
          ImageSize\  -> \ 
            500, 
AxesLabel\  -> \ {"r'", 
              I "s'"}, \ 
          PlotLabel\  -> \ "Gap

", \ 
PlotStyle\  -> \ {Thickness[0.005], Hue[ .3]}];\)\)

3.4.3.4   y-axis

\(\(ty\  = 
        Table[{\ zy\  = \ I\ gd\ 0.05\ k, \ 
            t\  /. \ 
FindRoot[fz[t]\  - \ zy\ , {t, I}] // 
              Chop[#, 10^\(-9\)] &}, {k, 0, 42}];\)\)

ly  =  
        Transpose[{Transpose[ty][[2]] // 
              Re, Transpose[ty][[2]] // Im}];

ly  =  Append[ly, {-1, 0}];

\(\(py\  = \ 
        ListPlot[ly, PlotJoined\  -> \ True, \ 
          ImageSize\  -> \ 
            500, 
AxesLabel\  -> \ {"r'", 
              I "s'"}, \ 
          PlotLabel\  -> \ "y-axis

", 
PlotStyle\  -> \ {Hue[0.9], 
              Thickness[ .005]}];\)\)

fz[-1]

tdo  =  -0.075;
ptp  =  
        Show[py, pg,  
          PlotRange  ->  {{-3, 2},  {- .2, 1.3}}, 
          PlotLabel  ->  "t'-plane\n\n",  
          AspectRatio  ->  0.4, ImageSize  ->  550, 
          Epilog  ->  {Thickness[ .005],  
              Line[{{0, 0}, {ra, 0}}], 
Hue[0],  
              Line[{{-1, 0}, {0, 0}}], Hue[ .9], 
              Text["y-axis", {0.5, 1.1}],  
Hue[
                0.6],  Line[{{-1, 0}, {-ra, 
                    0}}], 
Hue[0.3], 
              Text["-p", {-vp, tdo}], 
              Text["-1/p", {-1/vp, 
                  tdo}], 
Text["gap", {- 
                      0.5 (vp  +  1/vp), 1.29}]}];

\(\(Show[cou, \ ImageSize\  -> \ 550, 
        Epilog\  -> {\ pzb, pze, pzh, tzb, tze, 
            tzh, 
Text["I", {\(-1.5\), 0.5}], 
            Text["II", {1.5, 
                0.5}], \ 
Text["III", {\(-1.5\), 1.5}], 
            Text["IV", {1.5, 1.5}], 
Thickness[0.003], 
            Green, \ Line[{{\(-g\)/2 +  .079, gd}, {g/2 - 0.079, 
                  gd}}], 
Hue[ .9], \ 
            Line[{{0, 0}, {0, 
                  2.5}}]}, 
TextStyle\  -> {FontSize\  -> \ 14}];\)\)
vtst  =  0.1;
\(\(grtp = \ 
        Graphics[{ptb, pte, pth, ttb, tte, 
            tth, 
Text["I", { .4,  .4}], 
            Text["II", {\(-1\), 
                1.31}], 
Text["II", {\(-2.2\), 1.31}], 
            Text["II", { .5, 
                1.31}], 
Text["y = 0, x < 0", { .5, \(- .08\)}], Text["x > 0", {1.3, \(- .08\)}], 
Text["III", {\(- .7\), 0.4}], 
            Text["IV", {\(-1.6\),  .5}], 
            Blue, 
Text["y = D_", {\(-vp\) -  .17, \(- .08\)}], Text["y = D+", {\(-1.5\), \(- .08\)}], 
            Red, 
Text["y = D_", {\(- .15\), \(- .08\)}], Text["y = D+", {\(- .7\), \(- .08\)}]
}];\)\)
Show[ptp,  grtp, 
        PlotRange  ->  {{-3, 1.5}, {- .2, 1.5}}, 
        ImageSize  ->  
          750, 
Ticks  -> {Range[-3, 1], 
            Range[-2, 12, 2]  .1} , 
        TextStyle  -> {FontSize  ->  14}];

The figures above show the mapping of various domains within and above the condensor 
to corresponding domains in the t'-plane.  In particular, the part of I belonging to large 
negative Re(z) is mapped to the immediate neighbourhood of   t' = 0.

3.5  	An approximate expression for t' = (Subscript[f, 3]^(-1))(z) 
	valid for large negative z

For large gaps and  large negative real parts of z within the condensor  (0 ≤ y ≤ D) the numeric inversion fails 
since FindRoot[] no longer converges and sends error messages indicating irrecoverable accuracy loss.
Therefore it is necessary to find an approximate formulae for  (Subscript[f, 3]^(-1))(z) valid in this case. 

The part  x <  - g, 0 ≤ y ≤ D is mapped to a domain of the t'-plane in the neighbourhood of  t' = 0.
The trouble results from a cancellation between the then large logarithm term and the second term, 
which is large due to a large gap width g,  in the following  expression (eq.(2) rewritten with  
α = (((1 - p))^2)/p):

		z = (Subscript[f, 3])(t')  =  (D/π)[ ln t' + α((1/2)- (1/(1+t')) ) ] ,
		
The smallness of  |t'|  permits an iterative procedure. The equation above is rewritten as 
(t is used in place of t')

		π z/D := \(TraditionalForm\`z\&^\)  =   ln t  + α/2 - α/(1 + t)
		
		 \(TraditionalForm\`z\&^\)  + α/2  := \(TraditionalForm\`z\&_\) = ln t  +  α  - α/(1 + t) =  \(ln\ t\  + \ α\ t - α\ t\^2 + α\ t\^3\  - \ \  ... \) 	(9)
		
For very small  |t|  the terms proportional to  α  at the rhs may be neglected and  t = Exp[\(TraditionalForm\`z\&_\)]
as a zeroth approximation. in the next step the term α t is taken into account, in the third step
α t - α (t^2) are taken.  This procedure is done in Mathematica as follows:

3.5.1	0-th approximation

eq0 =  \(z\&_\)  ==    Log[t]

s0  =  Solve[eq0, t] // Flatten

t0  =  t  /.  s0

3.5.2	1-st approximation

eq1 =\(z\&_\)   == Log[t]  +  α t

t is augemnted bz a samll corrextion term  τ1 :

sua1  =  t  ->  t0  +  τ1

hq1  = eq1  /.  sua1

The rhs is expanded w.r.t. the small term  τ1 :

Series[hq1[[2]], {τ1, 0, 1}] // Normal // 
      Simplify

eq1a = \(z\&_\)   == PowerExpand[%]

so1a  = Solve[eq1a, τ1] // Flatten

So the first order approximation to (Subscript[f, 3]^(-1))(z) can be computed from the 
substitution below. This is not done here.

s1a  =   sua1  /.  so1a

3.5.3	2-nd approximation

Eq.(6)  is approximated as:

\(eq2\  = 
      z\&_\  == \ \ Log[t]\  + \ α\ t - α\ t\^2\)

A second correction term,  τ2,  is introduced:

sua2  =  t  ->  t0   +  τ1 +  τ2

hq2  = eq2  /.  sua2

gq21  = Series[hq2[[2]], {τ1, 0, 2}] // Normal // 
        Simplify

gq22  =  (Series[gq21, {τ2, 0, 1}] // Normal)  /.  so1a // 
        PowerExpand

\(eq2a\  = \ z\&_\  == %\)

so2a  = Solve[eq2a, τ2] // Flatten

s2a  =   (sua2  /.  so1a ) /.  so2a

According to eq.(6)  \(z\&_\) must be replaced by   π z/D + α/2.  So we get theapproximate formula given in the appendix A2 of pap82a 
and  scia[z] to be  introduced in subsec.2.2.3 of a planned new version of notebook SplitCond.nb.

10.  References

10.1 Papers, Books and Websites

0.             http://www.itp.tu-graz.ac.at/~schnizer/Theory_of_ Chambers/Plane_Geometries/

1.             Th. Heubrandtner,B. Schnizer,  G. Schweitzer, 
	Simple Models for RPC Weighting Fields and Potentials.
    	Nucl. Instrum. Methods Phys. Res., A : 535 (2004) no.1-2, pp.454-457.
	SchzP-82.pdf

2.              St. Rossegger,B. Schnizer,  G. Schweitzer,
	Comparison of simple models for RPC weighting fields and potentials.
             (in preparation)	

3.             G. Wendt, Static Fields and Stationary Currents (in German).	
             Encyclopedia of Physics (Ed. S. Fluegge), vol.16 , in particular pp.71 - 79 .
              Springer, Berlin 1958.

4.            J.Van Bladel,Electromagnetic Fields,
	McGraw-HillBook Comp.,New York 1964.
	pp. 136 - 144.

5.              G. Schweitzer, Quickly available graphics of static electromagnetic field distributions 
	given by conformal maps using Mathematica. 
             Graz,Techn.Univ.,Dipl.-Thesis.,1994
	Revised   1997.  http://www.itp.tu-graz.ac.at/~schnizer/ Conformal_Maps.html

6.              H. Kober, Dictionary of Conformal Representations.
              Dover, 1957.