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

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

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

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

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