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(Solved): Legendre polynomials form a basis set for the function space in the interval [-1,1] with the orthogo ...
Legendre polynomials form a basis set for the function space in the interval [-1,1] with the orthogonality realiton;
P_m(x)P_n(x) dx= 2/(2n+1)
n,m
P_0(x))=1, P_1(x)=x, P_2(x)=(3x^2)-1, P_3(x)=((5x^3)-3x)/2, P_4(x)=(35x^4-30x^2 +3)/8
Find c_1 with expasion coeffcient
f(X)=5x^4+8x^2+9x= sum[n,4] c_n*p_n(x)
Legendre polynomials form a basis set for the function space in the interval with the orthogonality relation Using the above relation and the polynomials given below find the expansion coeffients for the function given below.
Expert Answer
SolutionWe need to identify the values of c1 such that f(x) may be written as a linear combination of the Legendre polynomials Pn(x) in order to obtain the expansion coefficients c1 for the function f(x) using the Legendre polynomials.Given that f(x) = 5x4 + 8x3 + 9x, we can write the expansion of f(x) as:f(x) = ? c?P?(x), where the sum goes from n = 0 to infinity.To find the expansion coefficients c?, we need to compute the inner product of f(x) with each of the Legendre polynomials and divide by the appropriate normalization factor.