Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
very dicult to determine the equation of this curve. We ®rst need the equation of
the line joining P and Q in the z plane. The parametric equations of the line
joining P and Q are given by
x …2†
1 …2† ˆ y 1 ˆ t or x ˆ 3t 2; y ˆ 1 4t:
3 1
The equation of the line PQ is then given by z ˆ 3t 2 ‡ i…1 4t†. The curve in
the w plane into which the line PQ is mapped has the equation
w ˆ z 2 ˆ‰3t 2 ‡ i…1 4t†Š 2 ˆ 3 4t 7t 2 ‡ i…4 ‡ 22t 24t 2 †;
from which we obtain
u ˆ 3 4t 7t 2 ; v ˆ4 ‡ 22t 24t 2 :
By assigning various values to the parameter t, this curve may be graphed.
Sometimes it is convenient to superimpose the z and w planes. Then the images
of various points are located on the same plane and the function w ˆ f …z† may be
said to transform the complex plane to itself (or a part of itself).
Example 6.5
p
Map w ˆ f …z† ˆ z ; z ˆ re i :
p
Figure 6.4. The mapping function w ˆ z :
Solution: There are two square roots: f 1 …re i p
†ˆ r e i=2 p
; f 2 ˆf 1 ˆ r e i…‡2†=2 .
The function is double-valued, and the mapping is one-to-two. This is shown in
Fig. 6.4, where for simplicity we have used the same complex plane for both z and
w ˆ f …z†.
Branch lines and Riemann surfaces
p
We now take a close look at the function w ˆ z of Example 6.5. Suppose we allow z
to make a complete counterclockwise motion around the origin starting from point
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