Word Count : 616

---

We are asked to expand the function ( f(x) = x^4 - 1 ) in a series of the form ( \sum_{k=0}^{\infty} A_k P_k(x) ), where ( P_k(x) ) are Legendre polynomials. The standard approach is to use the orthogonality of Legendre polynomials over ([-1,1]) with weight 1: [ \int_{-1}^1 P_n(x) P_m(x) , dx = \frac{2}{2n+1} \delta_{nm}. ] The coefficients ( A_k ) are given by: [ A_k = \frac{2k+1}{2} \int_{-1}^1 f(x) P_k(x) , dx. ] We compute the first few coefficients. The Legendre polynomials are: [ P_0(x) = 1, \quad P_1(x) = x, \quad P_2(x) = \frac{3x^2-1}{2}, \quad P_3(x) = \frac{5x^3-3x}{2}, \quad P_4(x) = \frac{35x^4 - 30x^2 + 3}{8}. ] --- Step 1: Compute (A_0): [ A_0 = \frac{1}{2} \int_{-1}^1 (x^4 - ______ ___ _______ ______ ______ ______ ____ __________ _____ _________ ________ ____.
__________ ___ __________ ____ ___ __________ _______.
________ ____ ______ ______ _______.
________ _______ _____ ______ _____ ______ _________.
_____ ___ _________ ________ ______ ________ __________ ___.
_______ _________ _______ ____ __________.
_________ ___ ____ ______ _________ ____ __________ ________ ____.
_______ ____ _________ ________ ______ _________.
______ __________ __________ ______ ___ ________ __________ ______.
________ ____ __________ ___ _________ ________ _______ _______ _______ _______ _______ ________.
_________ ___ ____ _________ _____ _______.
____ ___ _________ ________ __________ ______.
___ _____ _________ ___ ___.
_______ ______ ______ _______ _______ ________.
_____ _________ _____ _____ ________.
_______ ____ _________ ____ __________.
__________ _____ _______ ______ ______ ____ ___ _____ _______ ______.
____ _____ _________ _____ _________ __________.
__________ _________ ___ _________ ______ ________ ________ ________ ____ __________.
________ _________ _________ __________ _______ ____ _____ _________ ___ ____.
___ ________ ________ _____ _________ ______.
____ __________ _____ ____ _______ _______.
_______ ______ _______ _______ _______.
__________ _________ _________ _____ ______ ____ ______.
_______ ____ _____ ___ __________ ______ ___ ____ ____ ______ _____ _______.
______ ____ __________ __________ ____.
______ ________ ________ __________ ________ __________.
_______ ________ ____ _________ ______ _______ _________ _________ _____.
________ ________ ____ ___ ____ __________ _________ ____ __________.
____ ____ ________ ___ ________ ______ __________ _____ ______ __________.
____ _____ _____ ______ __________ _____ _______ ______ _____ ____ _______.
__________ __________ ___ ______ _______ ____ __________ ______ __________ ______ _______ _____.
___ __________ ____ ___ ____ ____ ____ ________ ______ ________ __________.
_______ _________ __________ _______ ____ ___.
___ ____ ________ _______ ___ _______.
________ ________ ___ __________ _________ __________ _____.
_____ ____ ___ ________ ___ _____ ______ ___ __________.
__________ _______ ____ _____ ______.
___ ____ _______ __________ _______ ___ _________ _________ ________ _____ ________ _____.
_______ ______ ________ ____ _____.
________ ___ ___ ___ ___ _______ _________ ____.
_________ ______ ____ _____ _______ ___ _____.
__________ ____ _______ _____ ___ _______ ___ _________ _____ __________ ___.
______ ______ ____ ________ ______.
_______ ______ __________ _______ ________ ________ __________ __________ ____ _________ ____ ____.
___ ________ _____ ___ __________ _______ ______.
__________ ____ ______ ________ ___ __________ _________ __________ ______ ___ ___.
_______ __________ _____ ___ ________ ______ _______.
________ _____ ______ ___ _____ _____.
_______ __________ ________ _________ ______ _____ _____.
_________ __________ _________ _____ _____ ____ _____ ________ ________ _____ _______.
______ ________ __________ _________ _______ ___ ______ _________ __________.
_________ _______ ________ ___ ____ _________ ________ ______ ___.
__________ ________ _________ _____ _____ ____ ____ ____ _________ ___.
___ ____ _____ ______ ____ ________ ______.
________ ____ ____ ____ _________ _______ ______ ________.
______ _____ _________ ______ ____ ___ _________ ____ ________.
__________ _______ _________ _________ ___ ___ ________ _____.
_____ _______ _______ ___ ______ ________ ______ _____ ___.
__________ _________ ______ __________ ____ _____ _________ _______.
_______ _____ _______ _____ _______ ____ _____ ______ _________.
__________ ________ ________ _____ ________ _________ _________ ____.
Get Full Answer on WhatsApp