The expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines is known as a Fourier series. The orthogonality connections of the sine and cosine functions are used to create the Fourier series. Harmonic analysis is the study and computation of Fourier series, which is extremely useful for breaking down an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical.

Because the superposition principle holds for solutions of a linear homogeneous ordinary differential equation in the case of a single sinusoid, the solution for any arbitrary function can be found by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. This approach can even produce analytic answers in some exceptional circumstances when the Fourier series can be summed in closed form.

A generalized Fourier series equivalent to the Fourier series exists for any collection of functions that make up a full orthogonal system. A Fourier-Bessel series is created by exploiting orthogonality of the roots of a Bessel function of the first order.

Fourier Series and Integrals - Mathematics II

In this “Fourier Series and Integrals - Mathematics II” you will learn about the following topics:

  1. Introduction to Fourier series and integrals
  2. Periodic function and trigonometric series
  3. Fourier series
  4. Fourier sine and cosine series
  5. Fourier series in complex form
  6. Fourier integral
  7. Fourier Sine and Cosine integrals
  8. Fourier Sine and Cosine transforms.

==== Point to Note ====

This article Fourier Series and Integrals - Mathematics II is contributed by Namrata Chaudhary, a student of Lumbini Engineering College (LEC).

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