Quarter period

In mathematics, the quarter periods K(m) and iK ′(m) are special functions that appear in the theory of elliptic functions.

The quarter periods K and iK ′ are given by

and

When m is a real number, 0 ≤ m ≤ 1, then both K and K ′ are real numbers. By convention, K is called the real quarter period and iK ′ is called the imaginary quarter period. Any one of the numbers m, K, K ′, or K ′/K uniquely determines the others.

These functions appear in the theory of Jacobian elliptic functions; they are called quarter periods because the elliptic functions ${\displaystyle {\rm {sn}}u\,}$ and ${\displaystyle {\rm {cn}}u\,}$ are periodic functions with periods ${\displaystyle 4K\,}$ and ${\displaystyle 4{\rm {i}}K'\,}$ .

...
Wikipedia