**3** Fourier Series and Fourier Integrals

**3.1** Fourier Series

Let *f* (*x*) be a continuous, integrable function defined on the interval [–*c*, *c*]. Consider the Fourier series of *f* (*x*), *viz.*,

(3.1) |

The coefficients, *a*_{k} and *b*_{k}, indexed by the integers *k*, can be identified as follows. Multiply each side of (3.1) by cos(*n**π* *x**/**c*), *n* being an integer, and integrate over [–*c*, *c*] to obtain

To proceed, we note that

(3.2) |

where

(3.3) |

Thus,

(3.4) |

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