Legendre polynomials. Use the Gram–Schmidt procedure (Problem A.4) to orthonormalize the functions 1, x,x², and x³, on the interval -1 ≤ x ≤ 1. You may recognize the results—they are (apart from normalization) Legendre polynomials (Problem 2.64 and Table 4.1).

(Problem A.4)

Problem 2.64

Legendre’s differential equation reads

where ℓ is some (non-negative) real number.

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orthonormal. Problem A.4 Suppose you start out with a basis (le1), le2),..., len)) that is not The Gram-Schmidt procedure is a systematic ritual for generating from it an orthonormal basis (le), le),..., len)). It goes like this: (i) Normalize the first basis vector (divide by its norm): lei) = le₁) llei || (ii) Find the projection of the second vector along the first, and subtract it off: le₂) - (e₁|e2) lei). This vector is orthogonal tole); normalize it to get le₂). (iii) Subtract from le3) its projections alongle) and le): les) -(ei e3) |e₁)-(e₂|e3)|e₂). This is orthogonal to le) and le); normalize it to get leg). And so on. Use the Gram-Schmidt procedure to orthonormalize the 3-space basis lei) = (1 + i) î+ (1) ĵ+ (i) k, le₂2) = (i) î + (3) ĵ + (1) k, le3) = (0) î + (28) ĵ + (0) k