Express 5x³+x in Legendre polynomials

Short answer

From P_1 = x and P_3 = (5x^3 - 3x)/2, solve for 5x^3 = 2P_3 + 3P_1. Then f(x) = 5x^3 + x = 2P_3 + 4P_1.

Express the following in terms of Legendre's Polynomials: f(x) = 5x³ + x.

Legendre polynomials: P₁ = x, P₃ = (5x³ − 3x)/2.

From P₃ = (5x³ − 3x)/2 we get 5x³ = 2P₃ + 3x = 2P₃ + 3P₁.

f(x) = 5x³ + x = (2P₃ + 3P₁) + P₁ = 2P₃ + 4P₁.

Answer: 5x³ + x = 2P₃(x) + 4P₁(x).

Check: 2(5x³ − 3x)/2 + 4x = 5x³ − 3x + 4x = 5x³ + x ✓.

(Alternatively, put f = c₀P₀ + c₁P₁ + c₂P₂ + c₃P₃ and compare coefficients: x³ gives c₃ = 2, x gives c₁ − 3 = 1 so c₁ = 4, and c₂ = c₀ = 0.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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