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Chapter 12: Q9-1P (page 562)

Expand the following functions in the Legendre series.
$F (x) = [\begin{matrix} - 1 & - 1 & \leq & x & 0 \\ 1 & 0 & \leq & x & 1 \end{matrix}]$

Short Answer

Expert verified

Step by step solution

Concept of Legendre polynomial

Simplify the expression

Substitute for the value of m and P0(x)

Substitute for the value of m and P1(x)

Substitute for the value of m and P2(x)

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Most popular questions from this chapter

To calculate the given system of equation. $\frac{d}{dx} [{(x)}^{p} J_{p} (x)] = {(x)}^{p} J_{p - 1} (x)$ .

Expand the following functions in Legendre series.

$f (x) = arcsinx$

Verify equations (10.3) and (10.4).

(10.3) : (1-x²) u"-2 (m+1) xu'+[l(l+1) - m(m+1)] u=0

(10.4) : (1-x²) (u')" -2 [(m+1)+1] x(u')'+ [l(l+1) - (m+1)(m+2)]u'=0

To study the approximations in the table, a computer plots on the same axes the given function together with its small approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agrees with the function for large . If the small approximation is not clear, plot it alone with the function over a small interval $N3(x)$ .

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" $y^{''} + (\frac{λ}{x} - \frac{1}{4} - \frac{l (l + 1)}{x^{2}}) y = 0$ .

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Expand the following functions in the Legendre series.F(x)=[-1-1≤x010≤x1]

Short Answer

Step by step solution

Concept of Legendre polynomial

Simplify the expression

Substitute for the value of m and P0(x)

Substitute for the value of m and P1(x)

Substitute for the value of m and P2(x)

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Most popular questions from this chapter

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Expand the following functions in the Legendre series.
$F (x) = [\begin{matrix} - 1 & - 1 & \leq & x & 0 \\ 1 & 0 & \leq & x & 1 \end{matrix}]$