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688 CHAPTER 19 Complex Numbers and Functions

which can be made as large as we like by choosing y or −y large. Similarly, sin(iy) can be
arbitrarily large in magnitude.
In view of this new behavior, we might ask about periods and zeros of the complex sine and
cosine. We claim that the extension of these functions to the complex plane does not bring any
new periods or zeros.

THEOREM 19.9

1. sin(z) = 0 if and only if z = nπ for some integer n.
2. cos(z) = 0 if and only if z = (2n + 1)π/2 for some integer n.
3. cos(z) and sin(z) are periodic with periods 2nπ for every nonzero integer n. Furthermore,
these are the only periods of these functions.

These follow by systematic use of the real and imaginary parts of cos(z) and sin(z).For
example, to find the zeros of sin(z),solve
sin(z) = sin(x)cosh(y) + i cos(x)sinh(y) = 0.

Then
sin(x)cosh(y) = 0 and cos(x)sinh(y) = 0.
From the first equation and the fact that cosh(y) = 0 for real y,wehavesin(x) = 0, which for
real x means that x = nπ for some integer n. From the second equation,
n
cos(x)sinh(y) = cos(nπ)sinh(y) = (−1) sinh(y) = 0.
But then sinh(y) = 0so y = 0. Therefore z = nπ, with n an integer, proving part (1) of
Theorem 19.9. Parts (2) and (3) are proved similarly.
The other trigonometric functions are defined in terms of sine and cosine in the usual way.
For example, tan(z) = sin(z)/cos(z) for cos(z) = 0.

SECTION 19.3 PROBLEMS

In each of Problems 1 through 10, write the function value 12. Find u and v such that e 1/z =u(x, y) +iv(x, y).Show
in the form a + bi. that u and v satisfy the Cauchy-Riemann equations.
1. e i 13. Find u and v such that ze = u(x, y) + iv(x, y).Show
z
2. sin(1 − 4i) that u and v satisfy the Cauchy-Riemann equations
wherever they are defined.
3. cos(3 + 2i)
2
14. Find u and v such that cos (z) = u(x, y) + iv(x, y).
4. tan(3i)
Show that u and v satisfy the Cauchy-Riemann equa-
5. e 5+2i tions wherever they are defined.
6. cot(1 − πi/4) z
15. Find all solutions of e = 2i.
2
7. sin (1 + i)
16. Derive the following identities.
8. cos(2 − i) − sin(2 − i)
(a) sin(z + w) = sin(z)cos(w) + cos(z)sin(w).
9. e πi/2
(b) cos(z + w) = cos(z)cos(w) − sin(z)sin(w).
i
10. sin(e )
z
17. Find all solutions of e =−2.
z 2
11. Determine u and v such that e = u(x, y) + iv(x, y).
Show that u and v satisfy the Cauchy-Riemann equa- 18. Find all solutions of sin(z) = i.
tions.
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October 15, 2010 18:5 THM/NEIL Page-688 27410_19_ch19_p667-694