# §22.13 Derivatives and Differential Equations

## §22.13(i) Derivatives

Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. (The modulus $k$ is suppressed throughout the table.)

For alternative, and symmetric, formulations of these results see Carlson (2004, 2006a).

## §22.13(ii) First-Order Differential Equations

 22.13.1 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{sn}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left(1-{\mathop{\mathrm{sn}\/}\nolimits^{2}}\left(z,k\right)% \right)\left(1-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{2}}\left(z,k\right)% \right),$ 22.13.2 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{cn}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle={\left(1-{\mathop{\mathrm{cn}\/}\nolimits^{2}}\left(z,k\right)% \right)}{\left({k^{\prime}}^{2}+k^{2}{\mathop{\mathrm{cn}\/}\nolimits^{2}}% \left(z,k\right)\right)},$ 22.13.3 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{dn}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left(1-{\mathop{\mathrm{dn}\/}\nolimits^{2}}\left(z,k\right)% \right)\left({\mathop{\mathrm{dn}\/}\nolimits^{2}}\left(z,k\right)-{k^{\prime}% }^{2}\right).$
 22.13.4 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{cd}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left(1-{\mathop{\mathrm{cd}\/}\nolimits^{2}}\left(z,k\right)% \right)\left(1-k^{2}{\mathop{\mathrm{cd}\/}\nolimits^{2}}\left(z,k\right)% \right),$ 22.13.5 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{sd}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle={\left(1-{k^{\prime}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{2}}% \left(z,k\right)\right)}{\left(1+k^{2}{\mathop{\mathrm{sd}\/}\nolimits^{2}}% \left(z,k\right)\right)},$ 22.13.6 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{nd}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left({\mathop{\mathrm{nd}\/}\nolimits^{2}}\left(z,k\right)-1% \right)\left(1-{k^{\prime}}^{2}{\mathop{\mathrm{nd}\/}\nolimits^{2}}\left(z,k% \right)\right),$ 22.13.7 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{dc}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left({\mathop{\mathrm{dc}\/}\nolimits^{2}}\left(z,k\right)-1% \right)\left({\mathop{\mathrm{dc}\/}\nolimits^{2}}\left(z,k\right)-k^{2}\right),$ 22.13.8 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{nc}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle={\left(k^{2}+{k^{\prime}}^{2}{\mathop{\mathrm{nc}\/}\nolimits^{2% }}\left(z,k\right)\right)}{\left({\mathop{\mathrm{nc}\/}\nolimits^{2}}\left(z,% k\right)-1\right)},$ 22.13.9 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{sc}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left(1+{\mathop{\mathrm{sc}\/}\nolimits^{2}}\left(z,k\right)% \right)\left(1+{k^{\prime}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{2}}\left(z,k% \right)\right),$ 22.13.10 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{ns}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left({\mathop{\mathrm{ns}\/}\nolimits^{2}}\left(z,k\right)-k^{2% }\right)\left({\mathop{\mathrm{ns}\/}\nolimits^{2}}\left(z,k\right)-1\right),$ 22.13.11 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{ds}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left({\mathop{\mathrm{ds}\/}\nolimits^{2}}\left(z,k\right)-{k^{% \prime}}^{2}\right)\left(k^{2}+{\mathop{\mathrm{ds}\/}\nolimits^{2}}\left(z,k% \right)\right),$ 22.13.12 $\displaystyle\left(\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{cs}\/}% \nolimits\left(z,k\right)\right)^{2}$ $\displaystyle=\left(1+{\mathop{\mathrm{cs}\/}\nolimits^{2}}\left(z,k\right)% \right)\left({k^{\prime}}^{2}+{\mathop{\mathrm{cs}\/}\nolimits^{2}}\left(z,k% \right)\right).$

For alternative, and symmetric, formulations of these results see Carlson (2006a).

## §22.13(iii) Second-Order Differential Equations

 22.13.13 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{sn}\/}% \nolimits\left(z,k\right)$ $\displaystyle=-(1+k^{2})\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)+2k^{2}% {\mathop{\mathrm{sn}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.14 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{cn}\/}% \nolimits\left(z,k\right)$ $\displaystyle=-({k^{\prime}}^{2}-k^{2})\mathop{\mathrm{cn}\/}\nolimits\left(z,% k\right)-2k^{2}{\mathop{\mathrm{cn}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.15 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{dn}\/}% \nolimits\left(z,k\right)$ $\displaystyle=(1+{k^{\prime}}^{2})\mathop{\mathrm{dn}\/}\nolimits\left(z,k% \right)-2{\mathop{\mathrm{dn}\/}\nolimits^{3}}\left(z,k\right).$
 22.13.16 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{cd}\/}% \nolimits\left(z,k\right)$ $\displaystyle=-(1+k^{2})\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)+2k^{2}% {\mathop{\mathrm{cd}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.17 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{sd}\/}% \nolimits\left(z,k\right)$ $\displaystyle=(k^{2}-{k^{\prime}}^{2})\mathop{\mathrm{sd}\/}\nolimits\left(z,k% \right)-2k^{2}{k^{\prime}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{3}}\left(z,k% \right),$ 22.13.18 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{nd}\/}% \nolimits\left(z,k\right)$ $\displaystyle=(1+{k^{\prime}}^{2})\mathop{\mathrm{nd}\/}\nolimits\left(z,k% \right)-2{k^{\prime}}^{2}{\mathop{\mathrm{nd}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.19 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{dc}\/}% \nolimits\left(z,k\right)$ $\displaystyle=-(1+k^{2})\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)+2{% \mathop{\mathrm{dc}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.20 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{nc}\/}% \nolimits\left(z,k\right)$ $\displaystyle=(k^{2}-{k^{\prime}}^{2})\mathop{\mathrm{nc}\/}\nolimits\left(z,k% \right)+2{k^{\prime}}^{2}{\mathop{\mathrm{nc}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.21 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{sc}\/}% \nolimits\left(z,k\right)$ $\displaystyle=(1+{k^{\prime}}^{2})\mathop{\mathrm{sc}\/}\nolimits\left(z,k% \right)+2{k^{\prime}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.22 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{ns}\/}% \nolimits\left(z,k\right)$ $\displaystyle=-(1+k^{2})\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)+2{% \mathop{\mathrm{ns}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.23 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{ds}\/}% \nolimits\left(z,k\right)$ $\displaystyle=(k^{2}-{k^{\prime}}^{2})\mathop{\mathrm{ds}\/}\nolimits\left(z,k% \right)+2{\mathop{\mathrm{ds}\/}\nolimits^{3}}\left(z,k\right),$ 22.13.24 $\displaystyle\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\mathop{\mathrm{cs}\/}% \nolimits\left(z,k\right)$ $\displaystyle=(1+{k^{\prime}}^{2})\mathop{\mathrm{cs}\/}\nolimits\left(z,k% \right)+2{\mathop{\mathrm{cs}\/}\nolimits^{3}}\left(z,k\right).$

For alternative, and symmetric, formulations of these results see Carlson (2006a).