Quarter period
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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
and
Note that 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. 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 complementary parameter
- k is called the elliptic modulus
- k' 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 q 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.
[edit] References
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0486-61272-4. See chapters 16 and 17.