Advanced Mathematics for Engineers and Scientists by Paul DuChateau

Author:Paul DuChateau
Language: eng
Format: epub
ISBN: 9780486141596
Publisher: INscribe Digital
Published: 2012-10-19T04:00:00+00:00

(7.9)

It follows easily that

(7.10)

For some applications the exponential form of the Fourier series is more convenient than the trigonometric form.

Example 7.3 Exponential Form of the Fourier Series

Consider the function defined on [−π, π] by

Then

where

Thus, we have (formally) the following exponential series for ƒ(x):

Since

for every integer m we can write

This last series is the trigonometric form of the Fourier series for ƒ(x), [note that since ƒ(x) is even, it is a cosine series]. This example illustrates the equivalence of the exponential and trigonometric forms of the series.

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