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Fourier Series: Definition, Examples

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Any periodic function can be represented by a Fourier Series— a sum (an infinite series) of sines and cosines:

f(x) = A0 a1 cos x + a2 cos 2x +… + b1 sin x + b2 sin 2x +…

Each term is a periodic function with period 2π. Therefore, the sum of the series also has a period of 2π. The period can be replaced by one of arbitrary length, with the only issue being that the formulas will become a little more complicated to work with (Jackson, 2004).


Fourier series are useful in a wide range of fields including acoustics, with representation of musical sounds as sums of waves of various frequencies (Nearing, 2020) or quantum mechanics, for the quantum wave function of a particle in a box.

Example of a Fourier Series Approximation

A Sawtooth function, with period 1 and peak-to-peak amplitude of 1 can be represented fairly well by the Fourier series
S = 0.5 – 0.31831*sin(t) – 0.159155*sin(2*t) – 0.106103*sin(3*t)- 0.0795775*sin(4*t)-0.0636620*sin(5*t) (Schmahl, 2020):
fourier series example

Fourier Series vs. Taylor Series

These two series are both infinite expansions, but otherwise they are very different.

  • The Taylor series rewrites a general function as an infinite series of powers,
  • A Fourier series rewrites a function as an infinite series of sines and cosines.

The Taylor series expands a function, analytic in the neighborhood of some point x = a. The coefficients in the series are found by taking successive derivatives at point a:
Fourier Series vs. Taylor Series

The Taylor series is restricted to functions which can be differentiated to any degree, while the Fourier series doesn’t have this restriction. Fourier series also don’t have to meet the condition of continuity, and can be discontinuous at any number of finite points (Lanczos, 2016).


References

Jackson, D. (2004). F, Series and Orthogonal Polynomials. Dover Publications; Illustrated edition.
Lanczos, C. (2016). Discourse on F. Series. SIAM-Society for Industrial and Applied Mathematics.
Nearing, J. (2020). Fourier Series. Retrieved November 13, 2020 from: http://www.physics.miami.edu/~nearing/mathmethods/fourier_series.pdf
Peacock, J. (2013). Fourier Analysis. Retrieved November 13, 2020 from: https://physics.csuchico.edu/buchholtz/301A/Fourier.pdf
Schmahl, E. (1999). Examples of Fourier Series. Retrieved November 13, 2020 from: https://hesperia.gsfc.nasa.gov/~schmahl/fourier_tutorial/node2.html

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Stephanie Glen. "Fourier Series: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/fourier-series-definition-examples/
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