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Chapter 12: Q9-1P (page 562)

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

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

Prove $J_{p} (x) J'_{- P} (x) - J_{- p} (x) J'_{p} (x) = - \frac{2}{蟺 x} s i n p 蟺$ as follows:

Write Bessel's equation (12.1) with $y = J_{p}$ and with $y = J_{- p}$ ; multiply the $J_{p}$ equation by $J_{- p}$ and the $J_{- p}$ equation by $J_{p}$ and subtract to get $\frac{d}{d x} ['' x (J_{p} {J_{- p}}^{'} - J_{- p} J_{p}^{'})^{''}]^{''} = 0$ . Then $J_{p} {J_{- p}}^{'} - J_{- p}^{'} J_{p}^{'} = \frac{c}{x}$ . To find , use equation for each of the four functions and pick out the $1 / x$ terms in the products.

Expand the following functions in Legendre series.

Verify the recursion relations $(22.24)$ as follows:

a) Differentiate $(22 .23)$ With respect toto get $h 桅 = (h 鈭� 1) (鈭� 桅 / 鈭� x)$ equate coefficients ofrole="math" localid="1654857725406" $h^{n + 1}$

b) Differentiate $(22 .23)$ with respect to to get ${(1 鈭� h)}^{2} (鈭� 桅 / 鈭� h) = (1 鈭� h 鈭� x) 桅$ equate coefficients of $h$

c) Combine (a) and (b) to get $x (鈭� 桅 / 鈭� x) + h 桅鈭� h (1 鈭� h) 鈭� 桅 / 鈭� h = 0$ . Substitute the series for $桅$ and equate coefficients of $h^{n}$

Verify by direct substitution that the text solution of equation (16.3) and your solutions in the problems above are correct. Also prove in general that the solution (16.2) given for (16.1) is correct. Hint: These are exercises in partial differentiation. To verify the solution (16.4) of (16.3), we would change variables from x,y to say z, u where $y = x^{1 / 2} u, u = J_{1 / 3} (z), z = 2 x^{3 / 2}$ , and show that if x,y satisfy then u , z satisfy, $(z^{2} - \frac{1}{9}) u + z \frac{du}{dz} + z^{2} \frac{d^{2} u}{{dz}^{2}} = 0$ .

To calculate the given system of equation.

$\frac{d}{dx} [{(x)}^{p} J_{p} (x)] = {(x)}^{p} J_{p - 1} (x)$

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Expand the following functions in the Legendre series.F(x)=[-1-1鈮�x010鈮�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.
$F (x) = [\begin{matrix} - 1 & - 1 & 鈮� & x & 0 \\ 1 & 0 & 鈮� & x & 1 \end{matrix}]$