§19.4 Derivatives and Differential Equations

§19.4(i) Derivatives

 19.4.1 $\displaystyle\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}$ $\displaystyle=\frac{E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)}{k{k^{% \prime}}^{2}},$ $\displaystyle\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))% }{\mathrm{d}k}$ $\displaystyle=kK\left(k\right),$
 19.4.2 $\displaystyle\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}$ $\displaystyle=\frac{E\left(k\right)-K\left(k\right)}{k},$ $\displaystyle\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}$ $\displaystyle=-\frac{kE\left(k\right)}{{k^{\prime}}^{2}},$
 19.4.3 $\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{% \mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-% E\left(k\right)}{k^{2}{k^{\prime}}^{2}},$
 19.4.4 $\frac{\partial\Pi\left(\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^% {2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{\prime}}^{2}\Pi\left(\alpha^{2},k% \right)).$
 19.4.5 $\frac{\partial F\left(\phi,k\right)}{\partial k}={\frac{E\left(\phi,k\right)-{% k^{\prime}}^{2}F\left(\phi,k\right)}{k{k^{\prime}}^{2}}-\frac{k\sin\phi\cos% \phi}{{k^{\prime}}^{2}\sqrt{1-k^{2}{\sin}^{2}\phi}}},$
 19.4.6 $\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F% \left(\phi,k\right)}{k},$
 19.4.7 $\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{% \prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k\right)-{k^{\prime}}^{2}\Pi% \left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi\cos\phi}{\sqrt{1-k^{2}{% \sin}^{2}\phi}}\right).$

§19.4(ii) Differential Equations

Let $D_{k}=\ifrac{\partial}{\partial k}$. Then

 19.4.8 $(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F\left(\phi,k\right)=\frac{-k% \sin\phi\cos\phi}{(1-k^{2}{\sin}^{2}\phi)^{3/2}},$
 19.4.9 $(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)E\left(\phi,k\right)=\frac% {k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin}^{2}\phi}}.$

If $\phi=\pi/2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. An analogous differential equation of third order for $\Pi\left(\phi,\alpha^{2},k\right)$ is given in Byrd and Friedman (1971, 118.03).