What is the Fourier series? Overview, formula, and the advantages.

 Overview of Fourier Series

The name “Fourier” came from the name of well-known French mathematician and physicist Joseph Fourier. Fourier series is very useful in engineering problems.

He developed the Fourier series which is now used in various problems related to vibrations and heat transfer (heat equation, which is a type of partial differential equation) etc.

Before moving toward the Fourier series, first, we will study the periodic function.

Periodic function

If the value of each ordinate f(x) repeats itself at equal intervals in the abscissa, then f(x) is called a periodic function.

When f(x) = f(x + T) = f(x + 2T) = f(x + 3T) = f(x + 4T) = ……….

Here T is known as the period of the function f(x).

 Fourier series

A series of the form-

Is called the Fourier series, where an and bn are the constants.

Some advantages of Fourier series

• A function which is discontinuous can be represented by the Fourier series.
• The Fourier series of a discontinuous function is not uniformly convergent at all points.
• Expansion of an oscillating function by Fourier series provides all modes of oscillation.

The constant terms can be determined by the following formulas-

• a₀ can be determined as-

• a_n can be determined as-

•  b_n can be determined as-

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Know more here https://en.wikipedia.org/wiki/Fourier_series

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