610 Chapter 8so that- dw =^2 7l'Z 'd X
w
Also,
Hence {8.19-8) becomesAn= 11 f(x + ib)e-2 ... in(x+ib) dx {8.19-9)
If the strip b 1 < Im z < b 2 contains the real axis we may take b = O,
and {8.19-8) reduces to
{8.19-10)Corollary 8.22 The expansion {8.19-7) can be written in the trigono-
metric formwhere
00
f ( z) = 1/ 2 ao + L (an cos 21l'nz + bn sin 21l'nz)
n=lan =21
1
f(x+ib)cos27rn(x+ib)dxbn=21
1
f(x+ib)sin27rn(x+ib)dxProof Because of its absolute convergence in S, the series in (8.19-7) can
be written as
f(z) =Ao+ (A1e2 ... iz + A-1e-2 ... iz) + ... + (Ane21l'inz + A-ne-2 ... inz) + ...
ButAne^2 ·1rinz + A-ne-^2 11"inz = An(cos 21l'nz + i sin27rnz)
+ A_n (cos 21l'nz - i sin 27rnz)by letting
=(An+ A-n) cos 21l'nz + i(An - A-n) sin27rnz
= an cos 21l'nz + b~ sin 27rnz
and