1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.4 • TRIGONOMETRIC AND HYPERBOLIC F UNCT IONS 181

y

)

2

I

-K

2

• 1
  -2


• 3

.1!. 2

~·=sint

x

Figure 5.7 Vertical segments mapped onto hyperbolas by w = sin(z).

10

show that the images of these vertical segments are hyperbolas in the uv plane,
as Figure 5.7 illustrates. In Chapter 10, we give a more detailed analysis of the

mapping w = sinz.

Figure 5.7 suggests one big difference between the real and complex sine
functions. The real sine has the property that lsin xi ~ 1 for all real x. In Figure
5.7, however, the modulus of the complex sine appears to be unbounded, which
is indeed the case. Using Identity (5-33) gives

lsin zl^2 = lsinzcoshy + icosxsinhyl^2

= sin^2 xcosh^2 y+cos^2 xsinh^2 y

= sin^2 x (cosh^2 y -sinh^2 y) + sinh^2 y (cos^2 x + sin^2 x).

The identities cosh^2 y - sinh^2 y = 1 and cos^2 x + sin^2 x = 1 then yield

lsin zl

2

= sin^2 x + s in h^2 y.

A similar derivation produces

lcoszl^2 = cos^2 x + sinh^2 y.

If we set z = xo + iy in Identity (5-36) and let y -+ oo, we get

lim lsin (xo + ·iy)l^2 = s in^2 xo + lim sinh^2 y = oo.

y -oo y -oo

(5-36)

(5-37)

As advertised, we have shown that sin z is not a bounded function; it is also
evident from Identity (5-37) that cos z is unbounded.
The periodic character of the trigonometric functions makes apparent that
any point in their ranges is actually the image of infinitely many points.