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
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