Need some help understanding Hermite polynomials for PDE 11. The Hermite polynomials, H„(x), satisfy the following:į. < Hn, Hm >=Le-**H,(x)H,m(x) dx = VT2"n! 8n,m:ii. H, (x) = 2nH,n-1(x).п-1ii. Ни+1(х) — 2хН, (х) — 2пНp-1(х).nп-1iv. H, (х) %3D (-1)"е**.dn(e~x*).dxnUsing these, show:b. xe-**H,„(x)H„(x) dx = VT2"-1n! [8m,n=1 + 2(n + 1)d,m,n+1]

To prove: ∫-∞∞xe-x2HnxHmxdx=π2n-1n![δm,n-1+2n+1δm,n+1] Proof: Take the LHS ∫-∞∞xe-x2HnxHmxdx and…

11. The Hermite polynomials, H,„(x), satisfy the following: į. < Hn, Hm >= Se-**H„(x)Hm(x) dx = \T2"n! 8,m: ii. Н, (х) — 2nHm-1 (x). n-: ii. Ни+1 (х) — 2хН, (х) — 2nНи-1 (х). iv. H, (x) = (–1)"e*² (e-x*). dn (e-**). dxn Using these, show: b. xe-**H„(x)Hm(x) dx = VT2"-1n! [8mn=1+ 2(n + 1)8m,n+1] -x² тп-1

11. The Hermite polynomials, H,„(x), satisfy the following: į. < Hn, Hm >= Se-**H„(x)Hm(x) dx = \T2"n! 8,m: ii. Н, (х) — 2nHm-1 (x). n-: ii. Ни+1 (х) — 2хН, (х) — 2nНи-1 (х). iv. H, (x) = (–1)"e² (e-x). dn (e-). dxn Using these, show: b. xe-H„(x)Hm(x) dx = VT2"-1n! [8mn=1+ 2(n + 1)8m,n+1] -x² тп-1

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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