In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In mathematics, several functions are important enough to deserve their own name. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...

The quarter periods K and iK' are given by

and

iK'(m) = iK(1 − m)

Note that when m is a real number, , 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. Note that 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 sn u and cn u are periodic functions with period 4K. In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...

Note that the quarter periods are essentially the elliptic integral of the first kind, by making the substitution k² = 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: In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler. ...

• m is called the parameter
• m₁ = 1 − m is called the complementary parameter
• k is called the elliptic modulus
• k' is called the complementary elliptic modulus, where k'² = m₁
• α the modular angle, where k = sinα
• π / 2 − α the complementary modular angle. Note that m₁ = sin²(π / 2 − α) = cos²α

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

and

where ns and dn Jacobian elliptic functions. In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...

The nome q is given by

The complementary nome is given by

The real quarter period can be expressed as a Lambert series involving the nome: A Lambert series, named after Johann Heinrich Lambert, is a series taking the form It can be resummed by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of with the constant function Since this last sum is a typical number-theortic sum...

Additional expansions and relations can be found on the page for elliptic integrals. In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler. ...

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . See chapters 16 and 17.