Trigonometric series

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In mathematics, a trigonometric series is any series of the form:

\frac{1}{2}A_{o}+\displaystyle\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx}).[1]

It is called a Fourier series when the terms An and Bn have the form:

A_{n}=\frac{1}{\pi}\displaystyle\int^{2 \pi}_0\! f(x) \cos{nx} \,dx\qquad (n=0,1,2, \dots)
B_{n}=\frac{1}{\pi}\displaystyle\int^{2 \pi}_0\! f(x) \sin{nx}\, dx\qquad (n=1,2,3, \dots)

where f is an integrable function.[1]

It is not that case that every trigonometric series is a Fourier Series. A particular question of interest is given a trigonometric series, for which values of x does the series converge.

[edit] References

  • "Trigonmetric Series" by A. Zygmund
  1. ^ a b Fourier Series and Orthogonal Functions By Harry F. Davis. Page 89
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