Quarter period

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

The quarter periods K and iK &prime; 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 &lt; m &lt; 1, then both K and K &prime; are real numbers. By convention, K is called the real quarter period and iK &prime; is called the imaginary quarter period. Any one of the numbers m, K, K &prime;, or K &prime;/K uniquely determines the others.

These functions appear in the theory of Jacobian elliptic functions; they are called quarter periods because the elliptic functions $$\operatorname{sn}u$$ and $$\operatorname{cn}u$$ are periodic functions with periods $$4K$$ and $$4{\rm{i}}K'.$$ However, the $$\operatorname{sn}$$ function is also periodic with a smaller period (in terms of the absolute value) than $$4\mathrm iK'$$, namely $$2\mathrm iK'$$.

Notation
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:
 * $$m$$ is called the parameter
 * $$m_1= 1-m$$ is called the complementary parameter
 * $$k$$ is called the elliptic modulus
 * $$k'$$ is called the complementary elliptic modulus, where $${k'}^2=m_1$$
 * $$\alpha$$ the modular angle, where $$k=\sin \alpha,$$
 * $$\frac{\pi}{2}-\alpha$$ the complementary modular angle. Note that
 * $$m_1=\sin^2\left(\frac{\pi}{2}-\alpha\right)=\cos^2 \alpha.$$

The elliptic modulus can be expressed in terms of the quarter periods as


 * $$k=\operatorname{ns} (K+{\rm{i}}K')$$

and


 * $$k'= \operatorname{dn} K$$

where $$\operatorname{ns}$$ and $$\operatorname{dn}$$ are 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:


 * $$K=\frac{\pi}{2} + 2\pi\sum_{n=1}^\infty \frac{q^n}{1+q^{2n}}.$$

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