Quarter period

Template:Short description 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

<math>K(m)=\int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt {1-m \sin^2 \theta}}</math>

and

<math>{\rm{i}}K'(m) = {\rm{i}}K(1-m).\,</math>

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 <math>\operatorname{sn}u</math> and <math>\operatorname{cn}u</math> are periodic functions with periods <math>4K</math> and <math>4{\rm{i}}K'.</math> However, the <math>\operatorname{sn}</math> function is also periodic with a smaller period (in terms of the absolute value) than <math>4\mathrm iK'</math>, namely <math>2\mathrm iK'</math>.

Notation

The quarter periods are essentially the elliptic integral of the first kind, by making the substitution <math>k^2=m</math>. In this case, one writes <math>K(k)\,</math> instead of <math>K(m)</math>, understanding the difference between the two depends notationally on whether <math>k</math> or <math>m</math> is used. This notational difference has spawned a terminology to go with it:

• <math>m</math> is called the parameter
• <math>m_1= 1-m</math> is called the complementary parameter
• <math>k</math> is called the elliptic modulus
• <math>k'</math> is called the complementary elliptic modulus, where <math>{k'}^2=m_1</math>
• <math>\alpha</math> the modular angle, where <math>k=\sin \alpha,</math>
• <math>\frac{\pi}{2}-\alpha</math> the complementary modular angle. Note that

<math>m_1=\sin^2\left(\frac{\pi}{2}-\alpha\right)=\cos^2 \alpha.</math>

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

<math>k=\operatorname{ns} (K+{\rm{i}}K')</math>

and

<math>k'= \operatorname{dn} K</math>

where <math>\operatorname{ns}</math> and <math>\operatorname{dn}</math> are Jacobian elliptic functions.

The nome <math>q\,</math> is given by

<math>q=e^{-\frac{\pi K'}{K}}.</math>

The complementary nome is given by

<math>q_1=e^{-\frac{\pi K}{K'}}.</math>

The real quarter period can be expressed as a Lambert series involving the nome:

<math>K=\frac{\pi}{2} + 2\pi\sum_{n=1}^\infty \frac{q^n}{1+q^{2n}}.</math>

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. Template:ISBN. See chapters 16 and 17.