1550251515-Classical_Complex_Analysis__Gonzalez_

606 Chapter^8

for all z E S. The constant w is called a period off. It is easy to see if w is a

period off, then kw (k = ±1, ±2 ... ) is also a period. For instance, we have

f(z + 2w) = f(z + w + w) = f(z + w) = f(z)

and if ( k - 1 )w ( k > 1) is a period, then kw is also a period, since

f(z +kw)= J(z + (k - l)•v + w) = f(z + (k - l)w) = f(z)

Also, -kw (k > 0) is again a period, for

f(z - kw)= f(z - kw+ kw)= f(z)

If the function f: D -+ C has period w in D, and if w z E D for every

z E D, then the function F(z) = f(wz) has period 1 in D' = w-i D =

{z': z' = z/w, z ED}, for

F(z + 1) = f [w(z + 1)] = J(wz + w) = f(wz) = F(z)

Examples

1. f ( z) = sin z has period 27r in C and F(z) = sin 27!"Z has period 1 in C.


2. f(z) = ez has period 27ri in C and F(z) = e^2 Triz has period 1 in C.


3. f(z) = tanz has period 7r in C - {z: z =^1 / 2 (2k + l)7r}, and F(z) =
   tan7rz has period 1 in C - {z: z = %(2k + 1)}.

Theorem 8.45 If f is analytic in the strip

S = {z: bi < Imz < b21}

and periodic with period 1, then f(z) has a Fourier exponential series

representation

+oo

J(z) = L Ane2Trinz (8.19-1)

n=-oo

where

An= 1i J(x + ib)e-2Trin(x+ib) dx (8.19-2)

and the constant b is such that b 1 < b < b 2. The expansion (8.19-1) is

absolutely convergent for any z in S, and uniformly convergent on S' =

{z: bi < c1 :S Imz :S c2 < b2}·

Proof The function

(8.19-3)

defines a mapping of the strip bi < Im z < b 2 onto the ring r2 < lw I < ri,

where ri = e-^2 Trbi and r2 = e-^2 Trb2 (Fig. 8.13). In fact, if we let z = x + iy,