Principles of Mathematics in Operations Research

184 13 Power Series and Special Functions

The right hand side of the above inequality tends to oo as R —)• oo. Hence,

3i?o B \P(z)\ > n if \z\ > Ro- Since \P\ is continuous on the closed disc with

center at the origin and radius RQ, it attains its minimum; i.e. 3ZQ 9 |P(^o)| —

We claim that [i = 0. If not, put Q(z) = pZ)- Then, Q is nonconstant

polynomial, Q(0) = 1, and |<5(z)| > 1, Vz. There is a smallest integer k,

1 < k < n such that

Q{z) = 1 + bkzk + ••• + bnzn, bk ^ 0.

By Theorem 13.5.1 (d), 9el3 eik0bk = - \bk\. If r > 0 and rk \bk\ < 1, we

have |l + bkrkeike\ = 1 - rk \bk\, so that

|Q(rei0)| < 1 - rfc[|6fc| - r \bk+1\ rn~k \bn\\.

For sufficiently small r, the expression in squared braces is positive; hence

|<9(reiS)| < 1, Contradiction. Thus, \i = 0 = P(z 0 ). D

13.6 Fourier Series

Definition 13.6.1 A trigonometric polynomial is a finite sum of the form

N

f(x) = ao + 2_j ian cos nx + bn sin nx), x 6 M,

n=l

where ao,a\,..., a^, b\,..., 6 AT € C One can rewrite

N

-iV

which is more convenient. It is clear that, every trigonometric polynomial is

periodic, with period 2ir.

Remark 13.6.2 J/rt € N, emx is the derivative of —.— which also has period

2n. Hence,

n = 0,

2TT J~ 1 0, n = ±l,±2,...

— / einx dr = l *'

i/iye multiply f(x) by e~lmx where m £ Z, then if we integrate, we have

1

r emx dx

for \m\ < N. Otherwise, \m\ > N, the integral above is zero.
Therefore, the trigonometric polynomial is real if and only if

C—n — Cxi, 7i — U,... , iV.