Quarter_period
Quarter period
Special function in the theory of elliptic functions
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 and are periodic functions with periods and However, the function is also periodic with a smaller period (in terms of the absolute value) than , namely .