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\begin{document}
\title{Notes}
\date{May 7, 2014}
\maketitle
\section*{final}
turn in to mel, math dept secretary by noon on thursday.
\section*{last time}
finding the $n^\text{th}$ derivative of $J_v(x)$. $J_v(x)$ is bessel function 1st kind, order $v$. Satisfies $z^2w''(z)+zw'(z)+(z^2-v^2)w(z)=0$. showed
\begin{enumerate}
\item
  $(b^2-a^2)\int{J_v(ar)J_v(br)r\,\mathrm{d}r}=r\left(a{J_v}'(ar)J_v(br)b-J_v(ar){J_v}'(br)\right)$
\item
$\int{(J_v(ar))^2r\,\mathrm{d}r}$
\end{enumerate}
\begin{align*}
  \int{J_v(ar)J_v(br)r\,\mathrm{d}r}&=\frac{r}{b+a}\frac{1}{b-a}\left(a{J_v}'(ar)J_v(br)-bJ_v(ar){J_v}'(br)\right)\\
  \int{J_v(ar)^2r\,\mathrm{d}r}&=\frac{r}{2a}\frac{\mathrm{d}}{\mathrm{d}b}\left(\text{ above }\right)\\
  &=\frac{r}{2a}\left[a{J_v}'(ar)r{J_v}'(ar)-J_v(ar){J_\nu}'(ar)-aJ_\nu(ar){J_\nu}''(ar)r\right]\\
  &=\frac{r}{2a}\left[ar({J_\nu}'(ar))^2-J_\nu(ar)\left(ar{J_\nu}''(ar)+{J_\nu}'(ar)\right)\right]\\
  &=\frac{r}{2a}\left[ar({J_\nu}'(ar))^2-(J_\nu(ar)^2(\frac{\nu^2-(ar)^2}{ar}\right]\\
  \intertext{book page 239}
  \int_0^1{J_0(k_{oi}r)^2r\,\mathrm{d}r}&=\frac{1}{2}{J_1}^2(k_{oi})\\
  \int_0^1{J_n(k_{nm}r)r\,\mathrm{d}r}&=\frac{1}{2}\left[({J_n}'(k_{nm})^2+(J_n(k_{nm}))^2\right]-\frac{n^2}{2{k_{nm}}^2}(J_{n}(k_{nm}))^2=\frac{1}{2}(J_n'(k_{nm}))^2\\
  \int_0^1{J_0(k_{0m}r)^2r\,\mathrm{d}r}&=\frac{1}{2}J_0'(k_{om})^2
\end{align*}
\end{document}
