1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

182 CHAPTER 5 • ELEMENTARY FUNCTIONS

• EXAMPLE 5.10 Find the values of z for which cos z = cosh 2.

Solution Starting with Identity (5-35), we write

cosz = cosxcosh y - isinxsinh y = cosh 2.

If we equate real and imaginary parts, then we get

cos x cosh y = cosh 2 and sin x sinh y = 0.

The equation sinxsinhy = 0 implies either that x = 7m, where n is an integer ,

or that y = 0. Using y = 0 in the equation cos x cosh y = cosh 2 leads to the

impossible situation cos x = ~;:~ ~ = cosh 2 > 1. Therefore, x = 11'n, where n is

an integer. Since cosh ·y ;:: 1 for all values of y , the term cos x in the equation

cos x cosh y = cosh 2 must also be positive. For this reason we eliminate t he odd

values of n and get x = 211'k, where k is an integer.

Finally, we solve the equation cos 211'k cosh y = cosh y = cosh 2 and use the

fact that cosh y is an even function to conclude that y = ±2. Therefore, the

solutions to the equation cosz = cosh 2 are z = 2nk ± 2i, where k is an integer.

The hyperbolic functions also have practical use in putting t he tangent func-

tion into t he Cartesian form u +iv. Using Definition 5.6 and Equations (5-33)

and (5-35), we have

. sin (x + i y)
tanz=tan(x+iy)= (. )
cos x + i y

sin x cosh y + i cos x sinh y

cosx cosh y -isinxsinhy ·

If we multiply each term on the right by t he conjugate of the denominator,

the simplified result is

cosxsin x + icosh ysinh y

tanz=.

cos^2 x cosh^2 y + sin^2 x sinh^2 y

(5-38)

We leave it as an exercise to show that the ident ities cosh^2 y - sinh^2 y = 1 and

sinh 2y = 2 cosh y sinh y can be used in simplifying Equation (5-38) to get

sin 2x. sinh 2y

tanz = + i------

cos 2x + cosh 2y cos 2x + cosh 2y

(5-39)

As with sin z, we obtain a graph of the mapping w = tan z parametrically.

Consider the vertical line segments in the z plane obtained by successively setting

x = 7 + ~; fork= 0, 1,... , 8 and for each z value letting y vary continuously,

-3 ~ y ~ 3. In the exercises we ask you to show that the images of these vertical

segments are circular arcs in t he uv plane, as Figure 5.8 shows. In Chapter 10 ,

we give a more detailed investigation of the mapping w = tan z.