1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.1 • THE COMPLEX EXPONENTIAL FUNCTION 157

Note that parts (ii) and (iii) of the Theorem 5.1 combine to verify DeMoivre's
fonnula, which we introduced in Section 1.5 (see Identity (1-40)).
ff z = x + iy, t hen from parts (ii) and (iii) we have

exp (z) = e• = e"+iy = e"eiy = e" (cosy+ isin y). (5-1)

Some texts start with Identity (5-1) as the definition for exp (z). In the exer-
cises, we show that this is a natural approach from the standpoint of differential
equations.
The notation exp (z) is preferred over e• in some situations. For example,
exp (i) = 1.22140275816 ... is the value of exp (z) when z =! a.nd equals

the positive fifth root of e = 2.71828182845904 .... Thus, the notation e~ is

ambiguous and might be interpreted as any of the complex fifth roots of the
number e that we discussed in Section 1.5:

et ::;:; 1.22 140275816 (cos

2

~k + isin

2

~k) , fork= 0, 1, ... , 4.

To prevent this confusion, we often use exp(z) to denote the single-valued
exponential function.
We now explore some additional properties of exp (z). Using Identity (5-1),
we ca.n easily establish that

ez+i 2 n1f = ez,

e• = 1,

e~1 = ez2,

for all z, provided n is an integer;

iff z = i2mr, where n is an integer; and

iff z2 = z 1 + i2mr for some integer n.

(5-2)

(5-3)

(5-4)

For example, because Identity (5-1) involves the periodic functions cosy a.nd
sin y , a.ny two points in the z plane that lie on the same vertical line with their
imaginary parts differing by an integral multiple of 27T are mapped onto the