Quarter period explained

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

	\pi
	2

K(m)=\int

	d\theta
	\sqrt{1-m\sin²\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

\operatorname{sn}u

and

\operatorname{cn}u

are periodic functions with periods

and

4{\rm{i}}K'.

However, the

\operatorname{sn}

function is also periodic with a smaller period (in terms of the absolute value) than

4iK'

, namely

2iK'

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

is used. This notational difference has spawned a terminology to go with it:

is called the parameter

m₁₌1-m

is called the complementary parameter

is called the elliptic modulus

is called the complementary elliptic modulus, where

	2=m
{k'}
	1

\alpha

the modular angle, where

k=\sin\alpha,

	\pi
	2

-\alpha

the complementary modular angle. Note that

2\left(	\pi
	2

1=\sin

-\alpha\right)=\cos²\alpha.

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

k=\operatorname{ns}(K+{\rm{i}}K')

and

k'=\operatorname{dn}K

where

\operatorname{ns}

and

\operatorname{dn}

are Jacobian elliptic functions.

The nome

is given by

-	\piK'
	K

q=e

The complementary nome is given by

-	\piK
	K'

1=e

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

K=	\pi
	2

	infty
2\pi\sum
	n=1

	qⁿ
	1+q²ⁿ

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. . See chapters 16 and 17.