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 $\operatorname {sn} u$ 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 $4\mathrm {i} K'$ , namely $2\mathrm {i} K'$ .

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=\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.

References

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

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