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)"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
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 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2Fb0fd0daf-3851-471d-b872-19715227d9ad%2Fesdwhg_processed.png&w=3840&q=75)
Transcribed Image Text:**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.
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