Quarter period

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

and

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

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

and

k'= \textrm{dn} K\,

where ns and dn 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[edit]

  • Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. See chapters 16 and 17.