1dnGiven the Rodriguez formula Pn(x) = 2n!(x²-1)", show that Pm (x) = (1-x²)m/² mm Pn(x)2″n! dx" 1)”,dmdxmis a solution of the associated Legendre eqn. (1 − x²)y" − 2xy' + [n(n + 1) − _m²/2]y = 0.Hint: show that the equation. can be obtained by differentiating the Legendre Equation(1 —-x²) z"x²)z" − 2xz' + n(n+1)z = 0m times w.r.t. x, followed by the substitution z = y(1-x²)-m/2.

We begin by differentiating the Legendre equation, (1-x^2) y'' - 2xy' + n(n+1) y = 0 m times with…

Given the Rodriguez formula P₁(x) = 2"n! dx" (x²-1)", show that Pm (x) = (1-x²)m/2 mm Pn(x) dm dam is a solution of the associated Legendre eqn. (1-x²)y" - 2xy' + [n(n + 1) — [m²]y = 0. Hint: show that the equation. can be obtained by differentiating the Legendre Equation (1-x²)z" - 2xz' + n(n+1)z = 0 m times w.r.t. x, followed by the substitution z = y(1-x²)-m/2.

Given the Rodriguez formula P₁(x) = 2"n! dx" (x²-1)", show that Pm (x) = (1-x²)m/2 mm Pn(x) dm dam is a solution of the associated Legendre eqn. (1-x²)y" - 2xy' + [n(n + 1) — [m²]y = 0. Hint: show that the equation. can be obtained by differentiating the Legendre Equation (1-x²)z" - 2xz' + n(n+1)z = 0 m times w.r.t. x, followed by the substitution z = y(1-x²)-m/2.

University Physics Volume 3

17th Edition

ISBN:9781938168185

Author:William Moebs, Jeff Sanny

Publisher:William Moebs, Jeff Sanny

Chapter7: Quantum Mechanics

Section: Chapter Questions

Problem 7.1CYU: Check Your Understanding If a=3+4i , what is the product a* a?

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