# 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

$K(m)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}$ and

${\rm {i}}K'(m)={\rm {i}}K(1-m).\,$ 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.

The quarter periods are essentially the elliptic integral of the first kind, by making the substitution $k^{2}=m\,$ . In this case, one writes $K(k)\,$ instead of $K(m)\,$ , understanding the difference between the two depends notationally on whether $k\,$ or $m\,$ is used. This notational difference has spawned a terminology to go with it:

$m_{1}=\sin ^{2}\left({\frac {\pi }{2}}-\alpha \right)=\cos ^{2}\alpha .\,\!$ The elliptic modulus can be expressed in terms of the quarter periods as

$k={\textrm {ns}}(K+{\rm {i}}K')\,\!$ and

$k'={\textrm {dn}}K\,$ where ns and dn Jacobian elliptic functions.

$q=e^{-{\frac {\pi K'}{K}}}.\,$ The complementary nome is given by

$q_{1}=e^{-{\frac {\pi K}{K'}}}.\,$ The real quarter period can be expressed as a Lambert series involving the nome:

$K={\frac {\pi }{2}}+2\pi \sum _{n=1}^{\infty }{\frac {q^{n}}{1+q^{2n}}}.\,$ Additional expansions and relations can be found on the page for elliptic integrals.