Page 125 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

COMPLEX VARIABLES AND THE LAPLACE INVERSION INTEGRAL 115

2 1/2 1+ i
2 1/8 1
16

FIGURE 7.4: The roots of (1 + i) 1/4

Hence, the four roots are as follows:

π π
(1 + i) 1/4 = 2 1/8 cos + i sin
16 16

π π π π
1/8
= 2 cos + + i sin +
16 2 16 2
π π

= 2 1/8 cos + π + i sin + π
16 16
π 3π π 3π
= 2 1/8 cos + + i sin +
16 2 16 2
The locations of the roots are shown in Fig. 7.4.
The natural logarithm can be defined by writing z = re iθ for −π ≤ θ< π and noting
that

ln z = lnr + iθ (7.15)

and since z is not affected by adding 2nπ to θ this expression can also be written as

ln z = lnr + i (θ + 2nπ) with n = 0, 1, 2,... (7.16)

When n = 0weobtainthe principal branch. All of the single valued branches are analytic
for r > 0and θ 0 <θ <θ 0 + 2π.

7.1.1 Limits and Differentiation of Complex Variables: Analytic Functions
Consider a function of a complex variable f (z). We generally write

f (z) = u(x, y) + iv(x, y)