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11. The Hermite polynomials, H„(x), satisfy the following: į. < Hn, Hm >= L e-x*Hn(x)Hm(x) dx = VT2"n! 8n,m: т ii. H,(x) = 2nHn-1(x). п-1 ii. Нn+1 (x) %3D 2хН, (х) — 2nНm-1 (х). iv. H„(x) = (-1)"e* (e-x*). Using these, show: a. H – 2xH + 2nHn = 0 [use properties ii. And iii. ]

11. The Hermite polynomials, H„(x), satisfy the following: į. < Hn, Hm >= L e-x*Hn(x)Hm(x) dx = VT2"n! 8n,m: т ii. H,(x) = 2nHn-1(x). п-1 ii. Нn+1 (x) %3D 2хН, (х) — 2nНm-1 (х). iv. H„(x) = (-1)"e* (e-x*). Using these, show: a. H – 2xH + 2nHn = 0 [use properties ii. And iii. ]

Advanced Engineering Mathematics
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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Need help understanding hermite polynomials question for PDE. Thanks

11. The Hermite polynomials, H,(x), satisfy the following: п i. < Hn, Hm >= Se-xH,(x)H,m(x) dx VTT2"n! 8n,m. т ii. Н# (х) —D 2nНп-1 (х). ii. Нn+1 (х) — 2хН, (х) — 2nНт-1 (х). п-1 iv. H, (x) = (-1)"e** (e-**). dxn Using these, show: а. Н# — 2хН + 2nH, — 0 [useе properties i. And ii. ]
Transcribed Image Text:11. The Hermite polynomials, H,(x), satisfy the following: п i. < Hn, Hm >= Se-xH,(x)H,m(x) dx VTT2"n! 8n,m. т ii. Н# (х) —D 2nНп-1 (х). ii. Нn+1 (х) — 2хН, (х) — 2nНт-1 (х). п-1 iv. H, (x) = (-1)"e** (e-**). dxn Using these, show: а. Н# — 2хН + 2nH, — 0 [useе properties i. And ii. ]
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