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

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

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 ${\displaystyle {\rm {sn}}u\,}$ and ${\displaystyle {\rm {cn}}u\,}$ are periodic functions with periods ${\displaystyle 4K\,}$ and ${\displaystyle 4{\rm {i}}K'\,}$ .

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

${\displaystyle k={\textrm {ns}}(K+{\rm {i}}K')\,\!}$

and

${\displaystyle k'={\textrm {dn}}K\,}$

where ns and dn 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.