それはさておき
\begin{align*}
\eqalign{
& p = \sqrt {1 - k^2 \sin ^2 \phi } \cr
& k^2 \sin ^2 \phi = 1 - p^2 ,\;\;\;\;\sin ^2 \phi = k^{ - 2} - k^{ - 2} p^2 \cr
& \frac{d}
{{d\phi }}p^m = - mk^2 p^{m - 2} \sin \phi \cos \phi \cr}
\end{align*}
\begin{align*}
\eqalign{
& \frac{d}
{{d\phi }}p^m \sin \phi \cos \phi = \left( { - mk^2 p^{m - 2} \sin \phi \cos \phi } \right)\sin \phi \cos \phi + p^m \left( {1 - 2\sin ^2 \phi } \right) \cr
& = \left( {1 - 2\sin ^2 \phi } \right)p^m - mp^{m - 2} k^2 \sin ^2 \phi \cos ^2 \phi \cr
& = \left( {1 - 2k^{ - 2} + 2k^{ - 2} p^2 } \right)p^m - m\left( {1 - p^2 } \right)\left( {1 - k^{ - 2} + k^{ - 2} p^2 } \right)p^{m - 2} \cr
& = \left( {1 - 2k^{ - 2} } \right)p^m + 2k^{ - 2} p^{m + 2} - m\left( {1 - k^{ - 2} + k^{ - 2} p^2 } \right)p^{m - 2} + m\left( {1 - k^{ - 2} + k^{ - 2} p^2 } \right)p^m \cr
& = \left( {1 - 2k^{ - 2} } \right)p^m + 2k^{ - 2} p^{m + 2} - m\left( {1 - k^{ - 2} } \right)p^{m - 2} - mk^{ - 2} p^m + m\left( {1 - k^{ - 2} } \right)p^m + mk^{ - 2} p^{m + 2} \cr
& = - m\left( {1 - k^{ - 2} } \right)p^{m - 2} + \left\{ {1 - 2k^{ - 2} + m\left( {1 - 2k^{ - 2} } \right)} \right\}p^m + \left( {m + 2} \right)k^{ - 2} p^{m + 2} \cr
& = m\left( {k^{ - 2} - 1} \right)p^{m - 2} + \left( {m + 1} \right)\left( {1 - 2k^{ - 2} } \right)p^m + \left( {m + 2} \right)k^{ - 2} p^{m + 2} \cr}
\end{align*}
$$
m\left( {1 - k^2 } \right)\int {p^{m - 2} d\phi } - \left( {m + 1} \right)\left( {2 - k^2 } \right)\int {p^m d\phi } + \left( {m + 2} \right)\int {p^{m + 2} d\phi } = p^m \sin \phi \cos \phi
$$
$$
m\left( {1 - k^2 } \right)\int_0^{\frac{\pi }
{2}} {p^{m - 2} d\phi } - \left( {m + 1} \right)\left( {2 - k^2 } \right)\int_0^{\frac{\pi }
{2}} {p^m d\phi } + \left( {m + 2} \right)\int_0^{\frac{\pi }
{2}} {p^{m + 2} d\phi } = 0
$$
これを次数の上げ下げに使う
\begin{align*}
\eqalign{
& - \left( {1 - k^2 } \right)\int_0^{\frac{\pi }
{2}} {p^{ - 3} d\phi } + \int_0^{\frac{\pi }
{2}} {pd\phi } = 0 \cr
& \int_0^{\frac{\pi }
{2}} {p^{ - 3} d\phi } = \frac{{E\left( k \right)}}
{{1 - k^2 }} \cr
& \frac{d}
{{dk}}p^{ - 1} = k^{ - 1} \left( {p^{ - 3} - p^{ - 1} } \right) \cr
& \frac{d}
{{dk}}K\left( k \right) = \frac{1}
{k}\left\{ {\frac{{E\left( k \right)}}
{{1 - k^2 }} - K\left( k \right)} \right\} = - \frac{{K\left( k \right)}}
{k} + \frac{{E\left( k \right)}}
{{k\left( {1 - k^2 } \right)}} \cr}
\end{align*}