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.

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