Question
Using Rodrigue’s formula, obtain expression for the Hermite polynomial H3(x) and show that .
Answer :
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Rodrigue’s formula for Hermite polynomials is given by: [ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right) ] For (n = 3), we have: [ H_3(x) = (-1)^3 e^{x^2} \frac{d^3}{dx^3} \left( e^{-x^2} \right) = - e^{x^2} \frac{d^3}{dx^3} \left( e^{-x^2} \right) ] Step 1: Compute derivatives of (e^{-x^2}). [ \frac{d}{dx} (e^{-x^2}) = -2x e^{-x^2} ] [ \frac{d^2}{dx^2} (e^{-x^2}) = \frac{d}{dx} (-2x e^{-x^2}) = -2 e^{-x^2} + 4x^2 e^{-x^2} = (4x^2 - 2) e^{-x^2} ] [ \frac{d^3}{dx^3} (e^{-x^2}) = \frac{d}{dx} \big( (4x^2 - 2) e^{-x^2} \big) ]
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