Question

Expand each of the following polynomials in a Legendre series. You should get the same results that you got by a different method in the corresponding problems in Section 5.

7x⁴-3x+1

Short answer

The given polynomial in a Legendre series can be expanded as follows:

7x⁴-3x+1=8/5 P₄(x)+4P₂(x)-3P₁(x)+12/5 P₀(x)

Step-by-step solution

Concept of Legendre series:

For finding the coefficients of going similar to Fourier series formulae:

f(x) = ∑_l=0^∞c_lP_l(x) …… (1)

To find the coefficients c_l, multiply with P_m(x)and integrate.

Because Legendre polynomials are orthogonal, all the integrals on right are 0 except the one contains c_mand you can evaluate it:

∫_-1¹ [P_m(x)] dx=2/2m+1

Calculation to expand the given polynomial   in Legendre series:

As it is given 7x²-3x+1,

P₄(x) = 1/8 (35x⁴-3-x²+3)

8 P₄(x) = (35x⁴-3-x²+3)

Simplify further:

P₂(x) = 1/2 (3x²-1)

2P₂(x) = (3x²-1)

20 P₂(x) = (30x²-10)

So,

8 P₄(x) = (35x⁴-20 P₂(x)-10+3)

8 P₄(x) +20 P₂(x)+7=35x⁴

1/5 [8P₄(x)+20P₂(x)+7]=7x⁴

Simplify further as follows:

P_l(x)=x

7x⁴-3x+1=8/5 P₄(x)+4P₂(x)-3P₁(x)+7/5 P₀(x)+P₀(x)

7x⁴-3x+1=8/5 P₄(x)+4P₂(x)-3P₁(x)+12/5 P₀(x)