162 CHAPTER 5 • ELEMENTARY FUNCTIONS
- Let n be a positive integer. Show that
(a) (expz)" =exp (nz).
(b) (•x;z)" = exp (-nz).00 •15. Show that I: e"" converges for Im (z) > 0.
"'=O
16. Generalize Example 5.1, where t he condition -7' < c < d :::; "' is replaced by
d - c < 27'. Illustrate what this means.
17. Use the fa.ct that exp (z^2 ) is analytic to show that e"''-^112 sin 2xy is a harmonic
function.
18. Show tbe following concerning the exponential map.(a) The image of the line {(x,y): x = t, y = 27' + t} , where -oo < t < oo
is a spiral.
(b) T he image of the first quadrant{(x, y): x > 0, y > O} is the region
{w: lwl > 1}.
(c) If a is a real constant, the horizontal strip {(x,y): a< y :=:;a+ 2 7r} is
mapped one-tcrone and onto t he nonzero complex numbers.
(d) The image of the vertical line segment {(x, y): x = 2, y = t}, where
~ < t <^1 ; is half a circle.
(e) The image of t he horizont al ray { (x, y) : x > 0, y = i} is a ray.19. Explain how the complex function e• and t he real function e" are different. How
are they similar?
20. Many texts give an alternative defini tion for exp (z), starting with Identity
(5-1) as the definition for f (z) = exp (z). Recall that this identity states that
exp(z) = exp(x+iy) = e" (cosy+isin y). T his exercise shows such a definition
is a nat ural approach in terms of differential equations. We start by requiring
f (z) to be the solution to an initial-value problem satisfying t hree conditions:
(1) f is entire; (2) /' (z) = f (z) for all z; and (3) f (0) = 1. Suppose that
f (z) = f (x + iy) = u (x, y) +iv (x, y) satisfies conditions (1), (2), and (3).(a) Use the result f' (z) = u.,. ( x, y )+iv, ( x, y ) and the requirement f' ( z) =
f (z) from condition (2) to show that u, (x, y) - u (x, y) = 0, for all
z = (x, y).
(b) Show that the result in part (a) implies that t,_ [u (x, y) e - • J = 0. This
means u (x, y ) e-• is const ant with respect to x, sou. (x , y) e - • = p(y),
where p (y) is a function of y alone.
(c) Using a similar procedure for v (x , y) , show we wind up getting a pair
of solutions u(x,y) = p(y)e" and v(x,y) = q(y)e", where p(y) and
q (y) are functions of y alone.
(d) Now use the Cauchy- Riemann equations to conclude from part (c) that
p(y) = q' (y) and p' (y) = -q(y).