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^{\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 ≤ 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. 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 k2 = 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 complementary parameter
  • k is called the elliptic modulus
  • k' is called the complementary elliptic modulus, where {k'}^2=m_1\,\!
  • o\!\varepsilon\,\! the modular angle, where k=\sin o\!\varepsilon\,\!
  • \frac{\pi}{2}-o\!\varepsilon\,\! the complementary modular angle. Note that
m_1=\sin^2\left(\frac{\pi}{2}-o\!\varepsilon\right)=\cos^2 o\!\varepsilon.\,\!

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 complementary 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.

[edit] References

  • Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0486-61272-4. See chapters 16 and 17.
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