Page 781 - Advanced_Engineering_Mathematics o'neil

23.1 Conformal Mappings 761

components. Second, and perhaps more important, we can sometimes use this sequence to build
conformal mappings between given domains.
To illustrate the first point just made, we will state a general result about the action of a
bilinear transformation. In this statement, the term line refers to a straight line in the plane.

THEOREM 23.2
A bilinear transformation maps any circle to a circle or line, and any line to a circle or line.

Proof We need only verify the theorem for each of the three basis types: translations, rota-
tion/magnifications, and inversions. Then the theorem will be true for any compositions of these
mappings.
It is obvious geometrically that a translation maps a circle to a circle and a line to a line.
Similarly, a rotation/magnification maps a circle to a circle (perhaps of different radius) and a
line to a line. The issue comes down to the effect of an inversion on a circle or line. Begin with
the fact that any circle or line in the plane is the graph of an equation
2
2
A(x + y ) + Bx + Cy + R = 0
with A, B, C, and R as real numbers. This graph is a circle if A = 0and alineif A = 0 and B
and C are not both zero. Let z = x + iy. This equation becomes
B C
2
A|z| + (z + z) + (z − z) + R = 0.
2 2i
Now let w = T (z) = 1/z, which is an inversion. The image in the w plane of the locus of this
equation is the locus of
1 B 1 1 C 1 1
A + + + − + R = 0.
|w| 2 2 w w 2i w w
2
Multiply this equation by ww (the same as |w| ) to obtain
B C
2
R|w| + (w + w) − (w − w) + A = 0.
2 2i
In the w-plane, this is the equation of a circle if R = 0 and a line if A = 0 and B and C are not
both zero. This proves the theorem.

As the proof shows, translations and rotation/magnifications actually map lines to lines and
circles to circles, while an inversion may map a circle to a circle or line and a line to a circle
or line.

EXAMPLE 23.7

We will examine the action of the inversion w = 1/z on the vertical line Re(z) = a = 0. This is
the line x = a in the x, y-plane, and it consists of all complex numbers z = a + iy. The image of
such a point under the inversion is
1 a y
w = = − i = u + iv.
2
z a + y 2 z + y 2
2
It is routine to check that
2

1 1
2
u − + v = .
2a 4a 2
The image of the line x = a is therefore the circle of radius 1/2a with center (1/2a,0).

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