Quarter period

Special function in the theory of elliptic functions

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 {\displaystyle K(m)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}}

and

i K ′ ( m ) = i K ( 1 − m ) . {\displaystyle {\rm {i}}K'(m)={\rm {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 sn ⁡ u {\displaystyle \operatorname {sn} u} and cn ⁡ u {\displaystyle \operatorname {cn} u} are periodic functions with periods 4 K {\displaystyle 4K} and 4 i K ′ . {\displaystyle 4{\rm {i}}K'.} However, the sn {\displaystyle \operatorname {sn} } function is also periodic with a smaller period (in terms of the absolute value) than 4 i K ′ {\displaystyle 4\mathrm {i} K'} , namely 2 i K ′ {\displaystyle 2\mathrm {i} K'} .

Notation

The quarter periods are essentially the elliptic integral of the first kind, by making the substitution k 2 = m {\displaystyle k^{2}=m} . In this case, one writes K ( k ) {\displaystyle K(k)\,} instead of K ( m ) {\displaystyle K(m)} , understanding the difference between the two depends notationally on whether k {\displaystyle k} or m {\displaystyle m} is used. This notational difference has spawned a terminology to go with it:

  • m {\displaystyle m} is called the parameter
  • m 1 = 1 − m {\displaystyle m_{1}=1-m} is called the complementary parameter
  • k {\displaystyle k} is called the elliptic modulus
  • k ′ {\displaystyle k'} is called the complementary elliptic modulus, where k ′ 2 = m 1 {\displaystyle {k'}^{2}=m_{1}}
  • α {\displaystyle \alpha } the modular angle, where k = sin ⁡ α , {\displaystyle k=\sin \alpha ,}
  • π 2 − α {\displaystyle {\frac {\pi }{2}}-\alpha } the complementary modular angle. Note that
m 1 = sin 2 ⁡ ( π 2 − α ) = cos 2 ⁡ α . {\displaystyle m_{1}=\sin ^{2}\left({\frac {\pi }{2}}-\alpha \right)=\cos ^{2}\alpha .}

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

k = ns ⁡ ( K + i K ′ ) {\displaystyle k=\operatorname {ns} (K+{\rm {i}}K')}

and

k ′ = dn ⁡ K {\displaystyle k'=\operatorname {dn} K}

where ns {\displaystyle \operatorname {ns} } and dn {\displaystyle \operatorname {dn} } are Jacobian elliptic functions.

The nome q {\displaystyle q\,} is given by

q = e − π K ′ K . {\displaystyle q=e^{-{\frac {\pi K'}{K}}}.}

The complementary nome is given by

q 1 = e − π K K ′ . {\displaystyle q_{1}=e^{-{\frac {\pi 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 . {\displaystyle K={\frac {\pi }{2}}+2\pi \sum _{n=1}^{\infty }{\frac {q^{n}}{1+q^{2n}}}.}

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

References

  • Milton Abramowitz and Irene A. Stegun (1964), Handbook of Mathematical Functions, Dover Publications, New York. ISBN 0-486-61272-4. See chapters 16 and 17.