Quarter period

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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

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

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

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.

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