What are Fourier series?

Welcome back MechanicaLEi, did you know that Fourier series is named in honor of Jean-Baptiste 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 out the previous part of this series to learn about what orthogonal and orthonormal functions are? Now, let f of x be a two pi periodic function which is integrable on closed interval of minus pi and pi. Say, a not equals to one upon pi into integral of f of x dx from minus pi to pi, a n equals to one upon pi into integral of f of x into cos of n x dx from minus pi to pi and b n equals to one upon pi into integral of f of x into sine of n x dx from minus pi to pi. Then the trigonometric series a not upon two plus summation of a n into cos of n x plus b n into sine of n x from n equals to one to infinity is called the Fourier series of function f of x. Another type of Fourier series is a periodic function with period two into L. Suppose f is a periodic function with a period two L. Then the Fourier series of f is given by f of x equals to a not upon 2 plus summation of a n into cos of n pi x upon L plus b n into sine of n pi x upon L from n equals to one to infinity. Where the coefficient a not upon 2 is given by one upon 2 L into integral of f of x dx from minus L to L. Fourier cosine coefficient a m is given by one upon L into integral of f of x into co of n pi x upon L dx from minus L to L and finally Fourier sine coefficient, b n is given by one upon L into integral of f of x into sine 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 two L is?
In the next episode of MechanicaLEi find out what Dirichlet's conditions are?

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