1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions

From (8.19-9) it follows that

and

Also,

an= 1

1

f(x + ib)[e-21l"in(x-Hb) + e21l"in(x+ib)] dx

= 21

1

f(x + ib) cos 27rin(x + ib) dx

bn = i 1

1

f(x + ib)[e-21l"in(x+ib) _ e21l"in(x+ib)] dx

=21

1

f(x+ib)sin27rn(x+ib)dx

ao = 2Ao = 21

1

f(x + ib) dx

Therefore, we have

00

f ( z) =^1 / 2 ao + ·~::::can cos 27rnz + bn sin 27rnz)

n=l

with the coefficients an and bn given by (8.19-11) and (8.19-12).

611

(8.19-11)

(8.19-12)

(8.19-13)

Again, if the given strip contains the real axis, we may take b = 0 and
the formulas for the coefficients reduce to

an= 21

1

f(x)cos27rnxdx, bn = 21

1

f(x)sin27rnxdx

Example Let f(z) = tan7rz. This function is analytic and of period 1
in any horizontal strip that does not contain the real axis since the only
singularities of tan7rz are at the real points ±%(2k + 1), k = O, 1, 2, ....
We have

But

f(z) =

. e1l"iz _ e-1t"iz
-i emz ' + e-mz.
. e^2 7l"iz - 1. W - 1
=-i ' =-i--
e27l"iz + 1 w + 1

=F(w)

F(w) = -i (1--

2

-) = i(l - 2w + 2w^2 - 2w^3 + · · ·)

l+w

valid for lwl < 1. Hence we obtain

f(z) = i(l - 2e27l"iz + 2e47l"iz - 2e67l"iz + .. -)