Fourier analysis MOC

Fourier series

The Fourier series of a periodic function is an Infinite series of trigonometric functions, equal to the original function except at points of discontinuity.

Fourier series of a square wave. The above equation corresponds to .

Specifically, the Fourier series is an infinite series of cosine and sine functions at positive integer frequencies, so that

At points of discontinuity, for example the vertical sections in the square wave above, the value of the Fourier series is the average of the limits either side of the point. An alternate form is given by the Exponential Fourier series.

Finding the Fourier series

The following integrals yield values for and respectively¹

Properties

• It follows from sine and cosine being odd and even functions respectively that the Fourier series of an odd function has no cosine terms, and that of an even function has no sine terms. Additionally, note that for odd functions.
• The following orthogonality relations may prove useful where is the Kronecker delta.² Another useful property is
• (Fourier convergence theorem) Any piecewise differentiable function on the interval with period has a Fourier series convergent to where is continuous and the average of one-sided limits where is discontinuous.³

Relation to Fourier transform

The Fourier series may be thought of as a discrete version of the Fourier transform, which replaces summation of discrete frequencies with integration of a continuous range of frequencies.

Practice problems

• 2023. Advanced Mathematical Methods, p. 89 (§5 problems)
• 2017. Elementary differential equations and boundary value problems
  • introduction: pp. 476–477 (§10.2 problems)
  • convergence: pp. 481–482 (§10.3 problems)
  • odd/even: pp. 487–488 (§10.4 problems)

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