 Welcome back Mechanical AI! Did you know that several sets of orthogonal and orthonormal functions have become standard basis for approximating mathematical functions? This makes us wonder, what are orthogonal and orthonormal functions? Before we jump in, check the previous part of this series to learn about what residue theorem is. Now, two functions f of x and g of x are orthogonal over the closed interval of a comma b with neighboring function w of x if inner product of f of x and g of x which is equivalent to integral of f of x into g of x into w of x ds from a to b is equal to 0. If in addition the integral of f of x the whole square into w of x dx from a to b is equal to 1 and integral of g of x the whole square into w of x dx is equal to 1 also then the functions f of x and g of x are said to be orthonormal. Given an infinite orthogonal set consisting of values psi j from j equals 1 to infinity on closed interval of a comma b an orthogonal series expansion is summation of cj into psi j of x from j equals 1 to infinity where cj are constants. One main consideration is whether a given function f on closed interval of a comma b has an infinite orthogonal expansion that is f of x equals to summation of cj into psi j of x from j equals 1 to infinity for some constant cj which leads to Fourier transformation hence we first saw what orthogonal and orthonormal functions are and then went on to see what some of orthogonal series expressions are. 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 Fourier series are.