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

$K(m)={\int }_0^{\frac{\pi }{2}}\frac{d\theta }{\sqrt{1-m{\sin }^2\theta }}$

and

$\mathrm{i}{K}^{\prime }(m)=\mathrm{i}K(1-m).$

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 $\mathrm{sn}u$ and $\mathrm{cn}u$ are periodic functions with periods $4K$ and $4\mathrm{i}{K}^{\prime }.$ However, the $\mathrm{sn}$ function is also periodic with a smaller period (in terms of the absolute value) than $4\mathrm{i}{K}^{\prime }$, namely $2\mathrm{i}{K}^{\prime }$.

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
• $k^{\prime }$ is called the complementary elliptic modulus, where ${k^{\prime }}^2=m_1$
• $\alpha$ the modular angle, where $k=\sin \alpha ,$
• $\frac{\pi }{2}-\alpha$ the complementary modular angle. Note that

$m_1={\sin }^2(\frac{\pi }{2}-\alpha )={\cos }^2\alpha .$

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

$k=\mathrm{ns}(K+\mathrm{i}{K}^{\prime })$

and

$k^{\prime }=\mathrm{dn}K$

where $\mathrm{ns}$ and $\mathrm{dn}$ are Jacobian elliptic functions.

The nome $q$ is given by

$q=e^{-\frac{\pi K^{\prime }}{K}}.$

The complementary nome is given by

$q_1=e^{-\frac{\pi K}{K^{\prime }}}.$

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

$K=\frac{\pi }{2}+2\pi \sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{1+q^{2n}}.$

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. Template:ISBN. See chapters 16 and 17.