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Quarter period

Summary

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

${\displaystyle K(m)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}}$

and

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

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 ${\displaystyle \operatorname {sn} u}$ and ${\displaystyle \operatorname {cn} u}$ are periodic functions with periods ${\displaystyle 4K}$ and ${\displaystyle 4{\rm {i}}K'.}$ However, the ${\displaystyle \operatorname {sn} }$ function is also periodic with a smaller period (in terms of the absolute value) than ${\displaystyle 4\mathrm {i} K'}$, namely ${\displaystyle 2\mathrm {i} K'}$.

Notation

The quarter periods are essentially the elliptic integral of the first kind, by making the substitution ${\displaystyle k^{2}=m}$ . In this case, one writes ${\displaystyle K(k)\,}$  instead of ${\displaystyle K(m)}$ , understanding the difference between the two depends notationally on whether ${\displaystyle k}$  or ${\displaystyle m}$  is used. This notational difference has spawned a terminology to go with it:

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

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

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

and

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

where ${\displaystyle \operatorname {ns} }$  and ${\displaystyle \operatorname {dn} }$  are Jacobian elliptic functions.

The nome ${\displaystyle q\,}$  is given by

${\displaystyle q=e^{-{\frac {\pi K'}{K}}}.}$

The complementary nome is given by

${\displaystyle q_{1}=e^{-{\frac {\pi K}{K'}}}.}$

The real quarter period can be expressed as a Lambert series involving the nome:

${\displaystyle K={\frac {\pi }{2}}+2\pi \sum _{n=1}^{\infty }{\frac {q^{n}}{1+q^{2n}}}.}$

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.