612 Chapter^8
= i [1+2 f)-l)ne2'1rinzl
n=lvalid for e-^2 .,..iz < 1, or, Im z > 0.
Also, we may writeF( w) = -i (1 -!
1
) = -i (1 -! + 2_ -· · ·)
w 1+1/w w w2
valid for lwl > 1. Thus we obtain
f(z) = -i(l -2e-2.,..iz + 2e-4.,..iz - .. ·)
= -i [1+2 I)-1re-2.,..inzl
n=lvalid foir e_^2 .,..Y > 1, or, Im z < 0:
Exercises 8.8
Find Fourier exponential series for the following functions, valid in the
indicated regions.
1. f(z) = cot?rZ: (a) Imz > O; (b) Imz < 0
2. f(z) = sec2?rZ: (a) Imz > O; (b) Imz < 0
- f(z) = csc2?rZ: (a) Imz > O; (b) Imz < 0
4. f(z) = tanh7riz: (a) Imz > O; (b) Imz < 0
5. f(z) = sec^2 ?rZ: (a) Imz > O; (b) Imz < 0
- Show that An= J 01 f(x + ib)e-^2 .,..in(x+ib) dx is independent of b.
7. Let f be an entire function and suppose that (1) f is periodic with
period 1 in C, and (2) lf(z)I:::; Meclzl for some constants M > 0, c > 0.
Show that for some N,
N
f(z) = L Ame2.,..imz
m=-N8. Let f be analytic in the strip -b < Imz < b (b > 0), and suppose that
the series
+oo
F(z)= L f(z+n)
n=-oo
is uniformly convergent in every compact set contained in that strip