In mathematics, the quarter periods ''K''(''m'') and i''K'' ′(''m'') are

special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defin ...

s that appear in the theory of

elliptic functions In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those ...

. The quarter periods ''K'' and i''K'' ′ are given by :

K(m)=\int_0^ \frac

and :

K'(m) = 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 i''K'' ′ 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 In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tr ...

; they are called ''quarter periods'' because the elliptic functions

\operatornameu

and

\operatornameu

are periodic functions with periods

4K

and

4K'.

However, the

\operatorname

function is also periodic with a smaller period (in terms of the absolute value) than

4\mathrm iK'

, namely

2\mathrm iK'

Notation

The quarter periods are essentially the

elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...

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

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 In mathematics, the modular lambda function λ(τ)\lambda(\tau) is not a Modular form#Modular functions, modular function (per the Wikipedia definition), but every modular function is a rational function in \lambda(\tau). Some authors use a non-equ ...

k'

is called the complementary elliptic modulus, where

^2=m_1

\alpha

the

modular angle Angular eccentricity is one of many parameters which arise in the study of the ellipse or ellipsoid. It is denoted here by α (alpha). It may be defined in terms of the eccentricity, ''e'', or the aspect ratio, ''b/a'' (the ratio of the semi ...

, where

k=\sin \alpha,

\frac-\alpha

the complementary modular angle. Note that :

m_1=\sin^2\left(\frac-\alpha\right)=\cos^2 \alpha.

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

k=\operatorname (K+K')

and :

k'= \operatorname K

where

\operatorname

and

\operatorname

are

. The nome

q\,

is given by :

q=e^.

The complementary nome is given by :

q_1=e^.

The real quarter period can be expressed as a

Lambert series In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form :S(q)=\sum_^\infty a_n \frac . It can be resumed formally by expanding the denominator: :S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty ...

involving the nome: :

K=\frac + 2\pi\sum_^\infty \frac.

Additional expansions and relations can be found on the page for

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. Elliptic functions