# §19.4 Derivatives and Differential Equations

## §19.4(i) Derivatives

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

## §19.4(ii) Differential Equations

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

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

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