(a) The Rodrigues formula states that solves the Legendre equation d Pe(x) = = [ (1 – x²) dx е d 2²/01 (2²) ² ( ²² − 1)² (x² dx dPe (1) +l(l +1) P₂(x) = 0. Prove this is indeed the case by showing that the Rodrigues formula reproduces the recursion relation derived in class for the coefficients ak in the Taylor expansion P₁(x) = Σ akxk. dx (2)
(a) The Rodrigues formula states that solves the Legendre equation d Pe(x) = = [ (1 – x²) dx е d 2²/01 (2²) ² ( ²² − 1)² (x² dx dPe (1) +l(l +1) P₂(x) = 0. Prove this is indeed the case by showing that the Rodrigues formula reproduces the recursion relation derived in class for the coefficients ak in the Taylor expansion P₁(x) = Σ akxk. dx (2)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![(a) The Rodrigues formula states that
solves the Legendre equation
d
dx
1
Pi(x) = 2² (2) (x² - 1)
d
e
dx
(22
d
dx
(1 – x²). + l(l + 1) Pe(x) = 0.
dPe
dx
(b) Prove that the associate Legendre functions (m ≥ 0)
m
d
Pm(x) = (1 − x²)m/2 :) " Pe(x),
dx
(1
Prove this is indeed the case by showing that the Rodrigues formula reproduces the recursion
relation derived in class for the coefficients a in the Taylor expansion P₁(x) = Σ‰_。 ªkxk.
∞
k=0
(1)
solve the differential equation found at non-trivial m and l in the angular part of the Schrödinger
equation for spherically-symmetric potentials:
-x²) d/ pm (x)] + [e(l + 1). [2²]
dx
1 – x²
(2)
Pm (x) = 0.
(3)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3ebc26e-28a0-426e-bac7-61c517215e28%2F0b0760a0-b3c6-475e-a864-f636963f6ee3%2Fdh227w_processed.png&w=3840&q=75)
Transcribed Image Text:(a) The Rodrigues formula states that
solves the Legendre equation
d
dx
1
Pi(x) = 2² (2) (x² - 1)
d
e
dx
(22
d
dx
(1 – x²). + l(l + 1) Pe(x) = 0.
dPe
dx
(b) Prove that the associate Legendre functions (m ≥ 0)
m
d
Pm(x) = (1 − x²)m/2 :) " Pe(x),
dx
(1
Prove this is indeed the case by showing that the Rodrigues formula reproduces the recursion
relation derived in class for the coefficients a in the Taylor expansion P₁(x) = Σ‰_。 ªkxk.
∞
k=0
(1)
solve the differential equation found at non-trivial m and l in the angular part of the Schrödinger
equation for spherically-symmetric potentials:
-x²) d/ pm (x)] + [e(l + 1). [2²]
dx
1 – x²
(2)
Pm (x) = 0.
(3)
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