   Chapter 7.2, Problem 70E

Chapter
Section
Textbook Problem

# A finite Fourier series is given by the sum f ( x ) = ∑ n = 1 N a n sin n x          = a 1 sin x + a 2 sin 2 x + ... + a N sin N x Show that the mth coefficient a m is given by the formula a m = 1 π ∫ − π π f ( x )   sin m x   d x

To determine

To show: The mth coefficients am of Fourier Series is given by am=1πππf(x)sinmxdx.

Explanation

Finite Fourier series expansion of a function f(x) is given by the following sum:

f(x)=n=1Nansinnx …… (1)

Formula used:

The formula from Problem 68, ππsinmxsinnxdx={0     if mnπ     if m=n.

Given:

The finite Fourier Series:

f(x)=n=1Nansinnx

The expression for coefficient am to be proved

am=1πππf(x)sinmxdx

Proof:

Expand the summation notation given in equation (1):

f(x)=n=1Nansinnxf(x)=a1sinx+a2sin2x++aNsinNx …… (2)

Multiply on both sides with sinx and integrate each term from π to π:

ππf(x)sinxdx=ππa1sinxsinxdx+ππa2sin2xsinxdx++ππaNsinNxsinxdx

Recall that the integral of sinnxsinmx from π to π is non zero only when m=n

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