# Quarter period

In mathematics, the **quarter periods** *K*(*m*) and i*K* ′(*m*) are special functions that appear in the theory of elliptic functions.

The quarter periods *K* and i*K* ′ are given by

and

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 i*K* ′ 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 and are periodic functions with periods and .

The quarter periods are essentially the elliptic integral of the first kind, by making the substitution . In this case, one writes instead of , understanding the difference between the two depends notationally on whether or is used. This notational difference has spawned a terminology to go with it:

- is called the
**parameter** - is called the
**complementary parameter** - is called the
**elliptic modulus** - is called the
**complementary elliptic modulus**, where - the
**modular angle**, where - the
**complementary modular angle**. Note that

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

and

where ns and dn Jacobian elliptic functions.

The **nome** is given by

The **complementary nome** is given by

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

Additional expansions and relations can be found on the page for elliptic integrals.

## References

- 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.