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
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Need help understanding hermite polynomials question for PDE. Thanks
![**Title: Properties of Hermite Polynomials**
**11. The Hermite polynomials, \(H_n(x)\), satisfy the following:**
i. Orthogonality condition:
\[
\langle H_n, H_m \rangle = \int_{-\infty}^{\infty} e^{-x^2} H_n(x) H_m(x) \, dx = \sqrt{\pi} 2^n n! \, \delta_{n,m}.
\]
ii. Derivative property:
\[
H'_n(x) = 2nH_{n-1}(x).
\]
iii. Recurrence relation:
\[
H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x).
\]
iv. Explicit expression:
\[
H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} (e^{-x^2}).
\]
**Using these, show:**
a.
\[
H''_n - 2xH'_n + 2nH_n = 0
\]
[Use properties ii and iii.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2F92284dc4-b2d7-450c-8c44-38c817c00c0b%2Fcvxxju8_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Properties of Hermite Polynomials**
**11. The Hermite polynomials, \(H_n(x)\), satisfy the following:**
i. Orthogonality condition:
\[
\langle H_n, H_m \rangle = \int_{-\infty}^{\infty} e^{-x^2} H_n(x) H_m(x) \, dx = \sqrt{\pi} 2^n n! \, \delta_{n,m}.
\]
ii. Derivative property:
\[
H'_n(x) = 2nH_{n-1}(x).
\]
iii. Recurrence relation:
\[
H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x).
\]
iv. Explicit expression:
\[
H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} (e^{-x^2}).
\]
**Using these, show:**
a.
\[
H''_n - 2xH'_n + 2nH_n = 0
\]
[Use properties ii and iii.]
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