Mathematical Methods for Physicists: A concise introduction - Site Map

MAPPING
variable. If only one value of w corresponds to each value of z, we say that w is a
single-valued function of z or that f …z† is single-valued; and if more than one value
of w corresponds to each value of z, w is then a multiple-valued function of z. For
example, w ˆ z 2 p
is a single-valued function of z, but w ˆ z is a double-valued
function of z. In this chapter, whenever we speak of a function we shall mean a
single-valued function, unless otherwise stated.
Mapping
Note that w is also a complex variable and so can be written in the form
w ˆ u ‡ iv ˆ f …x ‡ iy†;
…6:7†
where u and v are real. By equating real and imaginary parts this is seen to be
equivalent to
u ˆ u…x; y†; v ˆ v…x; y†: …6:8†
If w ˆ f …z† is a single-valued function of z, then to each point of the complex z
plane, there corresponds a point in the complex w plane. If f …z† is multiple-valued,
a point in the z plane is mapped in general into more than one point. The
following two examples show the idea of mapping clearly.
Example 6.4
Map w ˆ z 2 ˆ r 2 e 2i :
Solution: This is single-valued function. The mapping is unique, but not
one-to-one. It is a two-to-one mapping, since z and z give the same square.
For example as shown in Fig. 6.3, z ˆ2 ‡ i and z ˆ 2 i are mapped to the
same point w ˆ 3 4i; andz ˆ 1 3i and 1 ‡ 3i are mapped into the same
point w ˆ8 6i.
The line joining the points P…2; 1† and Q…1; 3† in the z-plane is mapped by
w ˆ z 2 into a curve joining the image points P 0 …3; 4† and Q 0 …8; 6†. It is not
Figure 6.3. The mapping function w ˆ z 2 .
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