= Modulus derivative of the incomplete elliptic integral of the second kind
{title2=$\partial_k E(\phi,k)=[E(\phi,k)-F(\phi,k)]/k$}
For real $|k|<1$, define $E(\phi,k)=\int_0^\phi\sqrt{1-k^2\sin^2x}dx$ and $F(\phi,k)=\int_0^\phi(1-k^2\sin^2x)^{-1/2}dx$. Smoothness of the integrands permits <differentiation under the integral sign>. Subtracting the two integrands gives $E-F=-k^2\int_0^\phi\sin^2x/\sqrt{1-k^2\sin^2x}dx$, which is $k\partial_kE$. This proves the displayed identity for $k\ne0$. At $k=0$, both sides extend continuously with value zero. The variable $k$ is the modulus; derivatives with respect to the parameter $m=k^2$ instead obey $\partial_m E=(E-F)/(2m)$, with removable value $-\tfrac12\int_0^\phi\sin^2x\,dx$ at zero.
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