1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

10.4 • MAPPING BY TRIGONOMETRIC FUNCTIONS 419

)'

2

Z=exp(i2z~

Zi=-i

w = 1an z

v

J. -iZ+ i

/ w =--Z+I

x

Figure 10. 16 The composite transformation w = tan z.

for -oo < y < oo. Next, we rewrite these equations as

u

cosh y = -.- and

Sill a

Sill. h y = --. v

cos a

u

We now eliminate y from these equations by squaring and using the hyperbolic
identity cosh^2 y - sinh^2 y = 1. The result is the single equation
u2 v2
------ 1
sin^2 a cos^2 a - ·

(10-23)

T he curve given by Equation (10-23) is identified as a hyperbola in the uv plane
that has foci at the points (±1, 0). Therefore, the vertical line x = a is mapped
one-to-one onto the branch of the hyperbola given by Equation (10- 23 ) that

passes through the point (sina,O). If 0 <a< I• then it is the right branch;

if - 2 " < a < 0, it is the left branch. The image of the y-axis, which is the line

x = 0, is the v-ax.is. The images of several vertical lines are shown in Figure

10.l 7(a).

The image of the horizontal segment - 2 " < x < ~, y = b is the curve in the

w plane given by the parametric equat ions

u = sinxcoshb and v = cosxsinhb