Jacobi Elliptic Functions - Minor Functions

Minor Functions

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

$\begin{align} \operatorname{ns}(u) & = \frac{1}{\operatorname{sn}(u)} \\ \operatorname{nc}(u) & = \frac{1}{\operatorname{cn}(u)} \\ \operatorname{nd}(u) & = \frac{1}{\operatorname{dn}(u)} \end{align}$

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

$\begin{align} \operatorname{sc}(u) & = \frac{\operatorname{sn}(u)}{\operatorname{cn}(u)} \\ \operatorname{sd}(u) & = \frac{\operatorname{sn}(u)}{\operatorname{dn}(u)} \\ \operatorname{dc}(u) & = \frac{\operatorname{dn}(u)}{\operatorname{cn}(u)} \\ \operatorname{ds}(u) & = \frac{\operatorname{dn}(u)}{\operatorname{sn}(u)} \\ \operatorname{cs}(u) & = \frac{\operatorname{cn}(u)}{\operatorname{sn}(u)} \\ \operatorname{cd}(u) & = \frac{\operatorname{cn}(u)}{\operatorname{dn}(u)} \end{align}$

More compactly, we have

where each of p, q, and r is any of the letters s, c, d, n, with the understanding that ss = cc = dd = nn = 1.

(This notation is due to Gudermann and Glaisher and is not Jacobi's original notation.)

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