Quarter period

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

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 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 and are periodic functions with periods and .

NotationEdit

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.

ReferencesEdit

  • 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.