11. The Hermite polynomials, H, (x), satisfy the following: į. < Hn, Hm >= Se-** H„(x)H,m(x) dx = \T2"n! ,n,m: -x² ii. H, (x) = 2nH,-1(x). ii. Hu+1 (x) %3D 2x, (х) — 2nНn-1 (х). iv. H, (x) = (-1)"e** (e-*). an (e-**). dxn Using these, show: 0, с) Н, (0) п оdd, (2m)! [Let x = 0 in iii. And iterate п 3D 2m. т! Note from iv. That H, (x) = 1 and H,(x) = 2x.]
11. The Hermite polynomials, H, (x), satisfy the following: į. < Hn, Hm >= Se-** H„(x)H,m(x) dx = \T2"n! ,n,m: -x² ii. H, (x) = 2nH,-1(x). ii. Hu+1 (x) %3D 2x, (х) — 2nНn-1 (х). iv. H, (x) = (-1)"e** (e-*). an (e-**). dxn Using these, show: 0, с) Н, (0) п оdd, (2m)! [Let x = 0 in iii. And iterate п 3D 2m. т! Note from iv. That H, (x) = 1 and H,(x) = 2x.]
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 with c) hermite polynomials for PDE, thankyou..
![11. The Hermite polynomials, H, (x), satisfy the following:
i. < Hр, Нm >-
S e-x" H, (x)H,m(x) dx = \T2"n! 8n,m-
00
ii. H, (x) = 2nHn-1(x).
i. Ни+1 (х) — 2xНи (х) — 2nНn-1(х).
n+.
iv. H, (x) = (-1)"e*² d" (e-x*).
(e*).
dxn
Using these, show:
п оdd,
с) Н, (0) -
1(-1)m
(2m)!
[Let x = 0 in iii. And iterate
п 3 2m.
т!
Note from iv. That H, (x) = 1 and H1(x) = 2x.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2F704e3846-1653-45ba-9d92-d3882748b1b6%2Fp1smg6_processed.png&w=3840&q=75)
Transcribed Image Text:11. The Hermite polynomials, H, (x), satisfy the following:
i. < Hр, Нm >-
S e-x" H, (x)H,m(x) dx = \T2"n! 8n,m-
00
ii. H, (x) = 2nHn-1(x).
i. Ни+1 (х) — 2xНи (х) — 2nНn-1(х).
n+.
iv. H, (x) = (-1)"e*² d" (e-x*).
(e*).
dxn
Using these, show:
п оdd,
с) Н, (0) -
1(-1)m
(2m)!
[Let x = 0 in iii. And iterate
п 3 2m.
т!
Note from iv. That H, (x) = 1 and H1(x) = 2x.]
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