1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.1 • THE COMPLEX EXPONENTIAL FUNCTION 159

Thezplane.

w=el-

-+-

x

v

rj ..

Thew plane.

Figure 5.2 The fundamental period strip for the mapping w =exp (z).

Solving Equations (5-5) for x a.nd y yields

x = In p and y = <P + 2nn,

ri

(5-8)

where n is a.n integer. Thus, for any complex number w # 0, there are infinitely

many complex numbers z = x + iy such that w = ez. From Equations (5-8), the

numbers z a.re

z = x + iy = lnp + i (</! + 2nn)

= ln lwl+i(Argw+2nn),

where n is an integer. Hence

exp {ln lwl + i (Argw + 2nn)] = w.

(5-9)

In summary, the transformation w = e• maps the complex plane (infinitely

often) onto the set of nonzero complex numbers.
If we restrict the solutions to Equation (5-9) so that only the principal value
of the argument, - ?T < Argw ~ n, is used, the transformation w = e" = ex+iv
maps the horizontal strip {(x, y) : -?T < y ~ n} one-to-one and onto the range
set S = {w: w f O}. This strip is called the fundamental period strip and
is shown in Figure 5.2.
The horizontal line z = t + ib, for - oo < t < oo in the z plane, is mapped

onto the ray w = eteib = e' (cos b + i sin b) that is inclined at an angle <P = b in

thew plane. The vertical segment z =a+ iO, for - n < (} ~ n in the z plane,

is mapped onto the circle centered at the origin with radius e" in the w plane.

That is, w = e"eiO = e" (cosO + isinO). The lines r 1 , r2, and r3, are mapped to

the rays ri , r2, and r3, respectively. Likewise, the segments s1, s2, and s3 are

mapped to the corresponding circles si , s2, and s3.