The Hermite polynomials are orthogonal on the interval

Chapter 10, Problem 31E

(choose chapter or problem)

The Hermite polynomials \(H_{n}(x)\) are orthogonal on the interval \((-\infty, \infty)\)  with respect to the weight function \(W(x)=e^{-x^{2}}\). Verify this fact for the first three Hermite polynomials:

                         \(H_{0}(x) \equiv 1, \quad H_{1}(x)=2 x\)

                         \(H_{2}(x)=4 x^{2}-2\)

Equation Transcription:

Text Transcription:

H_n(x)

(-\infty, \infty)

W(x) = e^-x2

H_0(x)1,   H_1(x)=2x

H_2(x)=4x^2-2

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