Fourier series are expansions of Periodic Functions in terms of an infinite sum of
Sines and Cosines

(1) |

To compute a Fourier series, use the integral identities

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

where we have relabeled the term for future convenience but set and left for . Assume the function is periodic in the interval . Now use the orthogonality conditions to obtain

(8) |

(9) |

(10) |

Plugging back into the original series then gives

(11) | |||

(12) | |||

(13) |

for , 2, 3, .... The series expansion converges to the function (equal to the original function at points of continuity or to the average of the two limits at points of discontinuity)

(14) |

Near points of discontinuity, a ``ringing'' known as the Gibbs Phenomenon, illustrated above, occurs.
For a function periodic on an interval , use a change of variables to transform the interval of integration
to . Let

(15) | |||

(16) |

Solving for , . Plugging this in gives

(17) |

(18) |

Because the Sines and Cosines form a Complete Orthogonal Basis, the Superposition Principle holds, and the Fourier series of a linear combination of two functions is the same as the linear combination of the corresponding two series. The Coefficients for Fourier series expansions for a few common functions are given in Beyer (1987, pp. 411-412) and Byerly (1959, p. 51).

The notion of a Fourier series can also be extended to Complex Coefficients.
Consider a real-valued function . Write

(19) |

(20) |

(21) |

(22) |

For a function periodic in , these become

(23) |

(24) |

**References**

Arfken, G. ``Fourier Series.'' Ch. 14 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 760-793, 1985.

Beyer, W. H. (Ed.). *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, 1987.

Brown, J. W. and Churchill, R. V. *Fourier Series and Boundary Value Problems, 5th ed.* New York: McGraw-Hill, 1993.

Byerly, W. E. *An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.* New York: Dover, 1959.

Carslaw, H. S. *Introduction to the Theory of Fourier's Series and Integrals, 3rd ed., rev. and enl.*
New York: Dover, 1950.

Davis, H. F. *Fourier Series and Orthogonal Functions.* New York: Dover, 1963.

Dym, H. and McKean, H. P. *Fourier Series and Integrals.* New York: Academic Press, 1972.

Folland, G. B. *Fourier Analysis and Its Applications.* Pacific Grove, CA: Brooks/Cole, 1992.

Groemer, H. *Geometric Applications of Fourier Series and Spherical Harmonics.*
New York: Cambridge University Press, 1996.

Körner, T. W. *Fourier Analysis.* Cambridge, England: Cambridge University Press, 1988.

Körner, T. W. *Exercises for Fourier Analysis.* New York: Cambridge University Press, 1993.

Lighthill, M. J. *Introduction to Fourier Analysis and Generalised Functions.*
Cambridge, England: Cambridge University Press, 1958.

Morrison, N. *Introduction to Fourier Analysis.* New York: Wiley, 1994.

Sansone, G. ``Expansions in Fourier Series.'' Ch. 2 in *Orthogonal Functions, rev. English ed.*
New York: Dover, pp. 39-168, 1991.

© 1996-9

1999-05-26