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.

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 $4{\rm{i}}K'\,$ .

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=\textrm{ns} (K+{\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} + 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.

References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. See chapters 16 and 17.

This article is issued from Wikipedia - version of the 1/21/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.