Quarter period

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.

• 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.