1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

418 CHAPTER 10 • CONFORMAL MAPPING

15. Find the image of the sector r > 0, 0 < IJ < i• under w = !:;:.

16. Show that the function fi (z) in Equation (10-22) is analytic on the ray x < - 1,
    y =O.

10.4 Mapping by Trigonometric Functions

The trigonometric functions can be expressed with compositions that involve

the exponential function followed by a bilinear function. We can find images

of certain regions by following the shapes of successive images in the composite

mapping.

• EXAMPLE 10.1 2 Show that the transformation w = tan z is a one-to-one

conformal mapping of the vertical strip lxl < i onto the unit disk lwl < 1.

Solution Using Equations (5- 32 ) and (5- 34 ) , we write

1 ei• - e -iz -iei2z + i

w =tanz= i eiz + e -iz = ei2z + 1 ·

Then, mapping w = t an z can be considered to be the composition

-iZ + i ·2

w = and Z = e' Z.

Z + l

The function Z = exp(i2z) maps the vertical strip lxl < i one-to-one and onto

the right half-plane Re ( Z) > 0. Then the bilinear transformation w = -~~ii

maps the half-plane one-to-one and onto the disk, as shown in Figure 10.16.

• EXAMPLE 10.13 Show that the transformation w = f (z) = sinz is a one-

to-one conformal mapping. of the vertical strip lxl < ~ onto the w plane slit

along the rays u::; -1, v = 0, and u :'.:: 1, v = O.

Solution Because f' ( z ) = cos z =f 0 for values of z sat isfying the inequality

2" < Re(z) < ~. it follows that w = sinz is a conformal mapping. Using

Equation (5-33), we write

u +iv= sin z = sinx coshy + icosxsinhy.

If lal < ~' then the image of the vertical line x =a is the curve in the w plane

given by the parametric equations

u = sinacoshy and v = cosasinhy,