Quarter period - Alchetron, The Free Social Encyclopedia

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 ) = ∫ 0 π 2 d θ 1 − m sin 2 ⁡ θ

and

i K ′ ( m ) = i K ( 1 − m ) .

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 s n u and c n u are periodic functions with periods 4 K and 4 i K ′ .

Notation

The quarter periods are essentially the elliptic integral of the first kind, by making the substitution k 2 = 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

m 1 = 1 − m is called the complementary parameter

k is called the elliptic modulus

k ′ is called the complementary elliptic modulus, where k ′ 2 = m 1

α the modular angle, where k = sin ⁡ α

π 2 − α the complementary modular angle. Note that

m 1 = sin 2 ⁡ ( π 2 − α ) = cos 2 ⁡ α .

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

k = ns ( K + i K ′ )

and

k ′ = dn K

where ns and dn Jacobian elliptic functions.

The nome q is given by

q = e − π K ′ K .

The complementary nome is given by

q 1 = e − π K K ′ .

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

K = π 2 + 2 π ∑ n = 1 ∞ q n 1 + q 2 n .

Additional expansions and relations can be found on the page for elliptic integrals.

References

Quarter period Wikipedia

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