Fourier series is just the simplest type of series that can be used to expand some well-behaved functions. There exists a huge number of other series that can be used to expand well-behaved functions. In electromagnetics, for instance, when dealing with problems that exhibit spherical symmetry, one encounters the Legendre series (better known as the Fourier-Legendre series). Any well-behaved function f(x) can be expanded into a Legendre series in the interval -1<<1 using f(x) = Σan Pn(x), n=0 (1) where P(r) is the Legendre polynomial of order n, and the coefficients a,, are given by 2n+1 2 -1 1 f(x)P, (e) dr. Unlike Fourier series coefficients D., obtaining a,, is much harder than obtaining D., and most likely is subject to numerical integration procedures. (2) Write a program that calculates the coefficients defined in Eq. (2), plots a function f(z) versus zЄ [-1.1], and plots its Legendre series representation in Eq. (1) versus r€ (-1,1]. Obtain the plots when I =sin Tr (you may try other functions for your own if you like). A sample is shown below. f(x) = f(x) f₁(x) f₁(x) f10(x) -1.0 -0.5 0.0 0.5 1.0 X