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.)

Read more about this topic: Jacobi Elliptic Functions

Famous quotes containing the words minor and/or functions:

“People are too apt to treat God as if he were a minor royalty.”
—Herbert Beerbohm, Sir Tree (1853–1917)

“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (1903–1974)