Quarter period

In mathematics, the quarter periods K(m)\, and {\rm{i}}K'(m)\, are special functions that appear in the theory of elliptic functions.

The quarter periods K\, and {\rm{i}}K' \,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).\,

Note that when m is a real number, 0 \le m \le 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. Note that 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 {\rm{sn}} u\, and {\rm{cn}} u\, are periodic functions with periods 4K \, and 4{\rm{i}}K'\, .

Note that 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%2B{\rm{i}}K')\,\!

and

k'= \textrm{dn} K\,

where ns and dn Jacobian elliptic functions.

The nome q\, is given by

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} %2B 2\pi\sum_{n=1}^\infty \frac{q^n}{1%2Bq^{2n}}.\,

Additional expansions and relations can be found on the page for elliptic integrals.

References