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In [[mathematics]], the '''quarter periods''' ''K''(''m'') and i''K'' ′(''m'') are [[special function]]s that appear in the theory of [[elliptic functions]].
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The quarter periods ''K'' and i''K''&nbsp;&prime; are given by
 
:<math>K(m)=\int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt {1-m \sin^2 \theta}}</math>
 
and
 
:<math>{\rm{i}}K'(m) = {\rm{i}}K(1-m).\,</math>
 
When ''m'' is a real number, 0 &le; ''m'' &le; 1, then both ''K'' and ''K''&nbsp;&prime; are real numbers.  By convention, ''K'' is called the ''real quarter period'' and i''K''&nbsp;&prime; is called the ''imaginary quarter period''. Any one of the numbers ''m'', ''K'', ''K''&nbsp;&prime;, or ''K''&nbsp;&prime;/''K'' uniquely determines the others.
 
These functions appear in the theory of [[Jacobian elliptic functions]]; they are called ''quarter periods'' because the elliptic functions <math>{\rm{sn}} u\,</math> and <math> {\rm{cn}} u\,</math> are periodic functions with periods <math>4K \,</math> and <math>4{\rm{i}}K'\,</math> .
 
The quarter periods are essentially the [[elliptic integral]] of the first kind, by making the substitution <math>k^2=m\,</math>. In this case, one writes <math>K(k)\,</math> instead of <math>K(m)\,</math>, understanding the difference between the two depends notationally on whether <math>k\,</math> or <math>m\,</math> is used.  This notational difference has spawned a terminology to go with it:
* <math>m\,</math> is called the '''parameter'''
* <math>m_1= 1-m \,</math> is called the '''complementary parameter'''
* <math>k\,</math> is called the '''[[elliptic modulus]]'''
* <math>k' \,</math> is called the '''complementary elliptic modulus''', where <math>{k'}^2=m_1\,\!</math>
* <math>\alpha\,\!</math> the '''[[modular angle]]''', where <math>k=\sin \alpha\,\!</math>
* <math>\frac{\pi}{2}-\alpha\,\!</math> the '''complementary modular angle'''.  Note that
:<math>m_1=\sin^2\left(\frac{\pi}{2}-\alpha\right)=\cos^2 \alpha.\,\!</math>
 
The elliptic modulus can be expressed in terms of the quarter periods as
 
:<math>k=\textrm{ns} (K+{\rm{i}}K')\,\!</math>
 
and
 
:<math>k'= \textrm{dn} K\,</math>
 
where ns and dn [[Jacobian elliptic functions]].
 
The '''[[nome (mathematics)|nome]]''' <math>q\,</math> is given by
 
:<math>q=e^{-\frac{\pi K'}{K}}.\,</math>
 
The '''complementary nome''' is given by
 
:<math>q_1=e^{-\frac{\pi K}{K'}}.\,</math>
 
The real quarter period can be expressed as a [[Lambert series]] involving the nome:
 
:<math>K=\frac{\pi}{2} + 2\pi\sum_{n=1}^\infty \frac{q^n}{1+q^{2n}}.\,</math>
 
Additional expansions and relations can be found on the page for [[elliptic integral]]s.
 
==References==
* Milton Abramowitz and Irene A. Stegun, ''Handbook of Mathematical Functions'', (1964) Dover Publications, New York. ISBN 0-486-61272-4. See chapters 16 and 17.
 
[[Category:Elliptic functions]]

Latest revision as of 04:05, 5 December 2014

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