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## Fourier Series

More details about Fourier series can be found in Kreyszig, Chapter 10. Here we simply list some of the important results.

Firstly, if a function , defined over an interval , is extended as a periodic function so that

then may be expressed in terms of a Fourier series as
 (2.11)

with the Fourier coefficients of given by the Euler fromulae
 (2.12) (2.13) (2.14)

for positive integer values of . The derivation of theses coefficients requires some knowledge about the orthogonality properties of the trigonometric functions. The main properties that we need are given by the following integrals,

The proof of these integrals all follows the same pattern and requires the use of some trigonometric identities. For example, we may express

The integrand of the first term on the right hand side is always periodic over and so the integral will be zero. The second term on the right hand side is also periodic and will be zero provided that . The case gives the integral as

The other integrals are done in a similar manner, using

Next: Half-Range Expansions Up: Separation of Variables Previous: Separation of Variables
Prof. Alan Hood
2000-10-30