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

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.

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:

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

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

k = ns(K + iK')

and

k' = dnK

where ns and dn Jacobian elliptic functions.

The nome q is given by

q = exp( - πK' / K)

The complimentary nome is given by

q₁ = exp( - πK / K').

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 486-61272-4 . See chapters 16 and 17.