**Text Transcription for Educational Website:** 5. Use the recursion relation (5.2) to evaluate \(\int_{-1}^{1} x^m P_n(x) P_m(x) \, dx, n \le m\). 6. Expand the following in a Fourier-Legendre series for \(x \in (-1,1)\). a. \(f(x) = x^2\) b. \(f(x) = 5x^4 + 2x^3 - x + 3\) c. \(f(x) = \left\{\begin{array}{ll} -1, & -1 < x < 0, \\ 1, & 0 < x < 1. \end{array}\right.\) d. \(f(x) = \left\{\begin{array}{ll} x, & 0 < x < 1,\\ 0, & 0 < x < 1. \end{array}\right.\) Use integration by parts to show \(\Gamma(x+1) = x\Gamma(x)\). Prove the double factorial identities: \[ (2n-1)!! = \frac{(2n)!}{2^n n!} \] \[ (2n)!! = 2^n n! \]
**Text Transcription for Educational Website:** 5. Use the recursion relation (5.2) to evaluate \(\int_{-1}^{1} x^m P_n(x) P_m(x) \, dx, n \le m\). 6. Expand the following in a Fourier-Legendre series for \(x \in (-1,1)\). a. \(f(x) = x^2\) b. \(f(x) = 5x^4 + 2x^3 - x + 3\) c. \(f(x) = \left\{\begin{array}{ll} -1, & -1 < x < 0, \\ 1, & 0 < x < 1. \end{array}\right.\) d. \(f(x) = \left\{\begin{array}{ll} x, & 0 < x < 1,\\ 0, & 0 < x < 1. \end{array}\right.\) Use integration by parts to show \(\Gamma(x+1) = x\Gamma(x)\). Prove the double factorial identities: \[ (2n-1)!! = \frac{(2n)!}{2^n n!} \] \[ (2n)!! = 2^n n! \]
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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I need help with 6b
![**Text Transcription for Educational Website:**
5. Use the recursion relation (5.2) to evaluate \(\int_{-1}^{1} x^m P_n(x) P_m(x) \, dx, n \le m\).
6. Expand the following in a Fourier-Legendre series for \(x \in (-1,1)\).
a. \(f(x) = x^2\)
b. \(f(x) = 5x^4 + 2x^3 - x + 3\)
c. \(f(x) = \left\{\begin{array}{ll}
-1, & -1 < x < 0, \\
1, & 0 < x < 1.
\end{array}\right.\)
d. \(f(x) = \left\{\begin{array}{ll}
x, & 0 < x < 1,\\
0, & 0 < x < 1.
\end{array}\right.\)
Use integration by parts to show \(\Gamma(x+1) = x\Gamma(x)\).
Prove the double factorial identities:
\[
(2n-1)!! = \frac{(2n)!}{2^n n!}
\]
\[
(2n)!! = 2^n n!
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5f8fb56-c519-448a-bccd-239824e4adff%2Fb3446d07-bbc9-45af-9c2f-34bca5850a0a%2Fr5fvwol.jpeg&w=3840&q=75)
Transcribed Image Text:**Text Transcription for Educational Website:**
5. Use the recursion relation (5.2) to evaluate \(\int_{-1}^{1} x^m P_n(x) P_m(x) \, dx, n \le m\).
6. Expand the following in a Fourier-Legendre series for \(x \in (-1,1)\).
a. \(f(x) = x^2\)
b. \(f(x) = 5x^4 + 2x^3 - x + 3\)
c. \(f(x) = \left\{\begin{array}{ll}
-1, & -1 < x < 0, \\
1, & 0 < x < 1.
\end{array}\right.\)
d. \(f(x) = \left\{\begin{array}{ll}
x, & 0 < x < 1,\\
0, & 0 < x < 1.
\end{array}\right.\)
Use integration by parts to show \(\Gamma(x+1) = x\Gamma(x)\).
Prove the double factorial identities:
\[
(2n-1)!! = \frac{(2n)!}{2^n n!}
\]
\[
(2n)!! = 2^n n!
\]
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