Editing Jacobi elliptic functions (section)

=={{anchor|ns|nc|nd|sc|sd|dc|ds|cs|cd|pg}}Minor functions==

Reversing the order of the two letters of the function name results in the reciprocals of the three functions above:

:<math>
\operatorname{ns}(u) = \frac{1}{\operatorname{sn}(u)}, \qquad \operatorname{nc}(u)  = \frac{1}{\operatorname{cn}(u)},  \qquad 
\operatorname{nd}(u)  = \frac{1}{\operatorname{dn}(u)}.
</math>

Similarly, the ratios of the three primary functions correspond to the first letter of the numerator followed by the first letter of the denominator:

:<math>
\begin{align}
\operatorname{sc}(u)  = \frac{\operatorname{sn}(u)}{\operatorname{cn}(u)},  \qquad 
\operatorname{sd}(u)  = \frac{\operatorname{sn}(u)}{\operatorname{dn}(u)},  \qquad \operatorname{dc}(u)  = \frac{\operatorname{dn}(u)}{\operatorname{cn}(u)},  \qquad \operatorname{ds}(u)  = \frac{\operatorname{dn}(u)}{\operatorname{sn}(u)},  \qquad \operatorname{cs}(u)  = \frac{\operatorname{cn}(u)}{\operatorname{sn}(u)},  \qquad 
\operatorname{cd}(u) = \frac{\operatorname{cn}(u)}{\operatorname{dn}(u)}.
\end{align}
</math>
More compactly, we have

:<math>\operatorname{pq}(u)=\frac{\operatorname{pn}(u)}{\operatorname{qn}(u)}</math>

where p and q are any of the letters s, c, d.