91Ó°ÊÓ

Show that${\mathbf{J}}_{\mathbf{P}}\left(x\right)\mathbf{N}{\mathbf{\text{'}}}_{\mathbf{P}}\left(x\right)\mathbf{-}\mathbf{J}{\mathbf{\text{'}}}_{\mathbf{P}}\left(x\right){\mathbf{N}}_{\mathbf{P}}\left(x\right)\mathbf{=}\frac{\mathbf{J}{\mathbf{\text{'}}}_{\mathbf{P}}\left(x\right){\mathbf{J}}_{\mathbf{-}\mathbf{p}}\left(x\right)\mathbf{-}{\mathbf{J}}_{\mathbf{p}}\left(x\right)\mathbf{j}{\mathbf{\text{'}}}_{\mathbf{-}\mathbf{p}}\left(x\right)}{\mathbf{²õ¾\pm ²Ô\pm èÏ€}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{Ï€³\ae }}$.

Prove the given system of equations ${\mathit{I}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{√}\mathbf{\left(}\frac{\mathbf{2}}{\mathbf{π}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathbf{\right(}\mathit{x}\mathbf{)}$and${\mathit{K}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{√}\mathbf{(}\frac{\mathbf{π}}{\mathbf{2}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}{\mathit{e}}^{\mathbf{(}\mathbf{-}\mathbf{x}\mathbf{)}}\mathbf{.}$

To show ${\mathbf{J}}_{\mathbf{3}\mathbf{/}\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{x}}^{\mathbf{-}\mathbf{1}}{\mathbf{J}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.

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

P₁¹(³¦´Ç²õθ)

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

Expand the following functions in Legendre series.

Find the solutions of the following differential equations in terms of Bessel functions by using equations (16.1) and (16.2).

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{x}}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{(}\mathbf{4}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{)}\mathbf{y}\mathbf{=}\mathbf{0}$

Substitute the P_l(x), you found in Problems 4.3 or 5.3 into equation (10.6)to find P_l^mthen let x=cosθto evaluate:

P₄¹(³¦´Ç²õθ)

To sketch the graph of $\sqrt{\mathbf{x}}{\mathbf{J}}_{\frac{\mathbf{1}}{\mathbf{2}}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$for x from 0 to $\mathbf{Ï€}$.

We obtained (19.10) for${\mathit{J}}_{\mathbf{p}}\left(x\right)\mathbf{,}\mathit{p}\mathbf{â‰\yen }\mathbf{0}\mathbf{.}$It is, however, valid for$\mathit{p}\mathbf{â‰\yen }\mathbf{-}\mathbf{1}$, that is for${\mathit{N}}_{\mathbf{p}}\left(x\right)\mathbf{,}\mathbf{0}\mathbf{≤}\mathit{p}\mathbf{<}\mathbf{1}$. The difficulty in the proof occurs just after (19.7); we said that $\mathit{u}\mathbf{,}\mathit{v}\mathbf{,}\mathit{u}\mathbf{\text{'}}\mathbf{,}\mathit{v}\mathbf{\text{'}}$are finite at $\mathit{x}\mathbf{=}\mathbf{0}$which is not true for${\mathit{N}}_{\mathbf{p}}\left(x\right)$.

However, the negative powers of x cancel if$\mathit{p}\mathbf{<}\mathbf{1}$. Show this for $\mathit{p}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$by using two terms of the power series (12.9) or (13.1) for the function ${\mathit{N}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\left(x\right)\mathbf{=}\mathbf{-}{\mathit{J}}_{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}\left(x\right)$ [see (13.3)].