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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

K(m)=0π2dθ1msin2θ

and

iK(m)=iK(1m).

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 snu and cnu are periodic functions with periods 4K and 4iK .

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:

m1=sin2(π2α)=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=eπKK.

The complementary nome is given by

q1=eπKK.

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

K=π2+2πn=1qn1+q2n.

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.