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^{\pi/2} \frac{d\theta}{\sqrt {1-m \sin^2 \theta}}

and

iK'(m) = iK(1-m)

Note that when m is a real number,

0 \leq m \leq 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. Note that 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 sn u and cn u are periodic functions with period 4K. Note that 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
m₁ = 1-m is called the complimentary parameter
k is called the elliptic modulus
k' is called the complimentary elliptic modulus, where ${k'}^2=m_1$
$\alpha$ the modular angle, where $k=\sin \alpha$
$\pi/2-\alpha$ the complimentary modular angle. Note that $m_1=\sin^2 (\pi/2-\alpha)=\cos^2 \alpha$

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

k=\textrm{ns} (K+iK')

and

k'= \textrm{dn} K

where ns and dn Jacobian elliptic functions. The nome q is given by

q=\exp (-\pi K'/K)

The complimentary nome is given by

q_1=\exp (-\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, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . See chapters 16 and 17.

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