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Chapter 12: Q9-4P (page 562)

Expand the following functions in Legendre series.
$f (x) = arcsinx$

Short Answer

Expert verified

Step by step solution

01

Concept of Legendre polynomial

02

Simplify the expression

03

Substitute for the value of

$\begin{matrix} 2 c_{0} = \int_{- 1}^{1} \sin^{- 1} x P_{0} (x) d x \\ 2 c_{0} = {x \sin^{- 1} x - \int \frac{x}{\sqrt{1 - x^{2}}} d x}_{- 1}^{1} \\ c_{0} = 0 \end{matrix}$

04

Substitute for the value of m=1

05

Substitute for the value of m=2

06

Substitute for the value of m=3

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Most popular questions from this chapter

Show that d^l-m/dx^l-m (x²-1)^l=(l-m)!/(l+m)! (x²-1)^m d^l+m/dx^l+m (x²-1)^l.

Hint: Write(x²-1)^l=(x-1)^l(x+1)^land find the derivatives by Leibniz' rule.

Expand the following functions in Legendre series.

To show $N_{(2 n + 1)} (x) = {(- 1)}^{n + 1} J_{- (2 n + 1)} (x)$ .

To show $J_{3 / 2} (x) = x^{- 1} J_{1 / 2} (x)$ .

To study the approximations in the table, a computer plot on the same axes the given function together with its small x approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agreeing with the function for large x. If the small x approximation is not clear, plot it alone with the function over a small interval $j1(x)$

See all solutions

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