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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

Concept of Legendre polynomial

Simplify the expression

Substitute for the value of

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

Substitute for the value of m=1

Substitute for the value of m=2

Substitute for the value of m=3

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Most popular questions from this chapter

Solve the following eigenvalue problem (see end of Section 2 and problem 11): Given the differential equation $y^{''} + (\frac{λ}{x} - \frac{1}{4} - \frac{l (l + 1)}{x^{2}}) y = 0$ where $l$ is an integerlocalid="1654860659044" $\geq 0$ , find values of localid="1654860714122" $λ$ such that localid="1654860676211" $y \to 0$ aslocalid="1654860742759" role="math" $x \to \infty$ , and find the corresponding eigenfunctions. Hint: letlocalid="1654860764612" $y = x^{l + 1} e^{- x / 2} v (x)$ , and show that localid="1654860784518" $v (x)$ satisfies the differential equationlocalid="1654860800910" $x v^{''} + (2 l + 2 - x) v^{'} + (λ - l - 1) v = 0$ .Comparelocalid="1654860829619" $(22 .26)$ to show that if localid="1654860854431" $λ$ is an integerlocalid="1654860871428" $> l$ , there is a polynomial solution localid="1654860888067" $v (x) = L_{λ - t - 1}^{2 t + 1} (x)$ .Solve the eigenvalue problem localid="1654860910472" $y^{''} + (\frac{λ}{x} - \frac{1}{4} - \frac{l (l + 1)}{x^{2}}) y = 0$ .

(a) Using 15.2 , show that $\int_{0}^{\infty} J_{1} (x) dx = 1$ . (b) Use L23of the Laplace Transform Table (Page 469) to show that $\int_{0}^{\infty} J_{0} (x) dt = 1$ . (Also see Problem23.29.) $\frac{d}{dx} [x^{- p} J_{p} (x)] = - x^{- p} J_{p + 1} (x)$ .

Use problem 7 to show that

P_l^m (x) = (-1)^m (l+m)!/(l-m)! (1-x²)/2^l! d^l-m/dx^l-m(x²-1)^l

Use the recursion relation (5.8a) and the values of $P_{0} (x)$ and $P_{1} (x)$ to find localid="1664340078504" $P_{2} (x) P_{3} (x), P_{4} (x), P_{5} (x)$ , and $P_{6} (x)$ . [After you have found $P_{3} (x)$ , use it to find $P_{4} (x)$ and so on for the higher order polynomials.]

Solve $(22.9)$ to get $(22 .10)$ . If needed, see Chapter $(8)$ , Section 2. The given equation $(D + x) y_{0} = 0 toobtain y_{0} = e^{- \frac{x^{2}}{2}} .$

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

Short Answer

Step by step solution

Concept of Legendre polynomial

Simplify the expression

Substitute for the value of

Substitute for the value of m=1

Substitute for the value of m=2

Substitute for the value of m=3

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