91Ó°ÊÓ

Expand the following functions in Legendre series.

Show the spherical Bessel functions satisfy the differential equation ${\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{+}\mathbf{2}{\mathbf{xy}}^{\mathbf{\text{'}}}\mathbf{+}\left[x^2-n\left(n+1\right)\right]\mathbf{y}\mathbf{=}\mathbf{0}$.

To show ${\mathbf{N}}_{\left(2n+1\right)}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}\mathbf{+}\mathbf{1}}{\mathbf{J}}_{-\left(2n+1\right)}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.

Substitute the P₁(x)you found in Problems 4.3 or 5.3 into equation (10.6)to find, P_l^m(x); then let x=cos θto evaluate:

To show the following equation shown in the problem

$\mathbf{2}\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}{\mathit{J}}_{\mathbf{a}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{J}}_{\mathbf{a}\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$.

Show that ${\mathbf{∫}}_{\mathbf{-}\mathbf{1}}^{\mathbf{1}}{\mathbf{P}}_{\mathbf{m}}\left(x\right){\mathbf{P}}_{\mathbf{I}}\left(x\right)\mathbf{dx}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{if}\mathbf{}\mathbf{m}\mathbf{≤}\mathbf{I}$

To show that,$\underset{\mathbf{x}\mathbf{→}\mathbf{0}}{\mathbf{lim}}\frac{\mathbf{\left(}\mathbf{x}\mathbf{\right)}}{\mathbf{x}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$ .

(a) Using 15.2 , show that ${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{∞}}{\mathbf{J}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}\mathbf{=}\mathbf{1}$. (b) Use L23of the Laplace Transform Table (Page 469) to show that ${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{∞}}{\mathbf{J}}_0\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dt}\mathbf{=}\mathbf{1}$. (Also see Problem23.29.) $\frac{\mathbf{d}}{\mathbf{dx}}\left[x^{-p}{J}_p\left(x\right)\right]\mathbf{=}\mathbf{-}{\mathbf{x}}^{\mathbf{-}\mathbf{p}}{\mathbf{J}}_{\mathbf{p}\mathbf{+}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.