**3. Deriving the Legendre Polynomials**A general formula for the Legendre polynomials $ P_n(x) $ is given by\[P_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \left( x + \sqrt{x^2 - 1} \cos \theta \right)^n \, d\theta.\]Use the formula to determine $ P_1(x), P_2(x), $ and $ P_3(x) $.

The Legendre's Polynomial is Pnx=1π∫0πx+x2-1 cos θndθ ...... (1) To find the value of P1x ,…

Answered: 3. Deriving the Legendre Polynomials A…

3. Deriving the Legendre Polynomials A general formula for the Legendre polynomials P,(x) is given by 1 P.(2) = [ (2 + v7² - cos0)" d . Use the formula to determine P(x), P2(x), and P3(x).

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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**3. Deriving the Legendre Polynomials**

A general formula for the Legendre polynomials $ P_n(x) $ is given by

\[
P_n(x) = \frac{1}{\pi} \int_{0}^{\pi} \left( x + \sqrt{x^2 - 1} \cos \theta \right)^n \, d\theta.
\]

Use the formula to determine $ P_1(x), P_2(x), $ and $ P_3(x) $.

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