Need some help understanding Hermite polynomials for PDE **Hermite Polynomials and Their Properties**The Hermite polynomials, \( H_n(x) \), satisfy the following conditions:1. **Orthogonality Condition:** \[ \langle H_n, H_m \rangle = \int_{-\infty}^{\infty} e^{-x^2} H_n(x) H_m(x) \, dx = \sqrt{\pi} 2^n n! \, \delta_{n,m}. \] This expression represents the orthogonality of Hermite polynomials, where \(\delta_{n,m}\) is the Kronecker delta, which is 1 if \( n = m \) and 0 otherwise.2. **Derivative Relation:** \[ H'_n(x) = 2n H_{n-1}(x). \] This shows the relationship between the derivative of a Hermite polynomial and the polynomial of one lower order.3. **Recurrence Relation:** \[ H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x). \] This equation expresses the polynomial of order \( n+1 \) in terms of the polynomials of order \( n \) and \( n-1 \).4. **Explicit Formula:** \[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right). \] This formula provides a direct way to compute Hermite polynomials using differentiation.**Using These Properties:**Demonstrate the following integral involving Hermite polynomials:\[\int_{-\infty}^{\infty} x e^{-x^2} H_n(x) H_m(x) \, dx = \sqrt{\pi} 2^{n-1} n! \left[ \delta_{m,n-1} + 2(n+1) \delta_{m,n+1} \right].\]This expression illustrates how the integration of a product of a polynomial and a variable, weighted by a Gaussian function, results in specific coefficients that depend on the polynomial indices.

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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