It is possible at once to write down the most general solution for
and 
as governed by the equations of Art. 1.11.
This can be done by introducing the concept of functions of
a complex variable. If we let
and
then
represents the most general solution for and ,
where f is any differentiable function at all.
In any particular case, the problem is reduced to that of
finding a function f which will satisfy certain boundary
conditions.
The proof that Eq. (213) satisfies the basic conditions
on and expressed by Eqs. (208) and (209)
is straightforward. If we differentiate W with
respect to x we obtain
Differentiating W with respect to y, on the other hand, gives
Eliminating df/dZ between (214) and (215) yields
This equation, however, can only be satisfied if
and
where it is considered that the quantities and
are real. These equations are in accord with euations (208) and
(209) as can be seen by equating the two expressions for u and the
two expressions for v.
The conditions (217a,b) in fact represent the Cauchy-Riemann
conditions which are required if the function f(Z) is to
possess a unique derivative.
The relation between W and Z stipulated by (213) implies
that for each point x, y in the Z-plane there corresponds
at least one value of and in the W-plane.
One can map out the isolines of and in the
x, y plane or isolines of x and y in the 
plane.
In either case the sets of isolines are orthogonal. The
procedure of transferring from one plane to the other
constitutes conformal mapping.
Consider an arbitrary curve C on the Z-plane. This can
be mapped on the W-plane through a given transform
. Suppose the curve C and its mapped
counterpart are as shown in Fig. 1.11-2. Any point on
the curve in the Z-plane can be represented by x, y or by
the polar coordinates 
. The complex quantity
Z can be written in terms of r and 
as
The corresponding point in the W-plane can
be expressed by the polar coordinates 
. In terms
of these quantities the complex potential is
.
The solution of a problem of Laplacian flow in the vicinity
of a fixed boundary is tantamount to finding that
transform which maps the boundary as the
straight line = constant in the W-plane. That is, for
a fixed boundary the flow must be along the boundary and
consequently the latter must be a surface of constant .
Consider the problem of ascertaining the two-dimensional,
irrotational, divergenceless flow in the neighborhood of the
solid boundary formed by two planes intersecting at an arbitrary
angle 
(see Figure 1.11-3a).
The boundary, c - c', can be mapped on the W-plane as a straight line by use of the following transform:
If we introduce the complex polar form of W and Z given by Eqs. (218) and (219) we have from Eq. (220)
which implies that
and
If we then choose the value of n as
then the curve c - c' will represent the axis of on the
W plane, i.e. the line 
, which satisfies the
condition that the boundary is a streamline.
The streamlines for other positions in the fluid can now be mapped
from the W-plane back onto the Z-plane by use of the
relations (221a,b) and (222). As a specific example suppose
radians, then
and
For any particular isoline = constant, the value of
R consistent with a selected 
can be found from
the isoline of in the W-plane and mapped
on the Z-plane using the transformed coordinates
from Eqs. (223a,b).
The isolines of can be mapped on the Z-plane in a similar
manner. The
resulting orthogonal fields of and are shown
in Fig. 1.11-4. Either set of isolines determines the flow
uniquely in the neighborhood of the boundary. The isolines
of obtained in this illustration represent a family
of hyperbolas whose major axis is the isoline and
whose asymptotes are the boundary lines c - c'.
If we express W as the complex potential 
in
Eq. (220) then we find
whence
for this example. The velocity components 
and
are given by
for Laplacian flow in general. Thus, for this
Hence at point 0 the velocity is zero, i.e. the point of intersection of the straight portions of the boundary represents a stagnation point in the flow. Eqs. (226) indicate that the velocity increases with distance from the boundary, the magnitude simply being
Consequently, the solution can only apply over a finite distance from the boundary if the motion is to be physically real (i.e. have a finite velocity at infinity).
A thorough discussion of conformal transformations is beyond the scope of this text. For a more thorough treatment including a wide variety of solved problems of this type, reference is made to Milne-Thompson (1938).