 Welcome back MechanicalEI, did you know that the Fourier series named in honour of Gene baptised Joseph Fourier introduced the series for the purpose of solving the heat equation in a metal plate? This makes us wonder, what are Fourier series? Before we jump in, check the previous part of the series to learn about what orthogonal and orthonormal functions are. Now, let f of x be a 2 pi periodic function which is integrable on closed interval of minus pi and pi. Let a naught equals to 1 upon pi into integral of f of x dx from minus pi to pi. a n equals to 1 upon pi into integral of f of x into cos of nx dx from minus pi to pi. And b n equals to 1 upon pi into integral of f of x into sin of nx dx from minus pi to pi. Then the trigonometric series a naught upon 2 plus summation of a n into cos of nx plus b n into sin of nx from n equals 1 to infinity is called the Fourier series of function f of x. Another type of Fourier series is a periodic function with period 2 into l. Suppose f is a periodic function with a period 2 l, then the Fourier series of f is given by f of x equals to a naught upon 2 plus summation of a n into cos of n pi x upon l plus b n into sin of n pi x upon l from n equals 1 to infinity, where the coefficient a naught upon 2 is given by 1 upon 2 l into integral of f of x dx from minus l to l. Fourier cosine coefficient a m is given by 1 upon l into integral of f of x into cos of n pi x upon l dx from minus l to l. And finally Fourier sine coefficient b n is given by 1 upon l into integral of f of x into sin of n pi x upon l dx from minus l to l. Hence we first saw what Fourier series with period 2 pi are and then went on to see what Fourier series of period 2 l is. So like, subscribe and comment with your feedback to help us make better videos. Thanks for watching. Also thanks a lot for those constructive comments. You helped the channel grow. So here are the top mechanical EIs of our last videos. In the next episode of Mechanical EI, find out what Dirichlet's conditions are.