Quarter period

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. . See chapters 16 and 17.