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Cosine amplitude

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elliptic cosine

One of the three basic Jacobi elliptic functions, denoted by

The cosine amplitude is expressible in terms of the Weierstrass sigma-functions, the Jacobi theta-functions or a power series, as follows:

\mathop{\rm cn} u = \ \mathop{\rm cn} ( u, k) = \ \frac{\sigma _ {1} ( u) }{\sigma _ {3} ( u) } = \ \frac{\theta _ {0} ( 0) \theta _ {2} ( \upsilon ) }{\theta _ {2} ( 0) \theta _ {0} ( \upsilon ) } =

= \ 1 - \frac{u ^ {2} }{2! } + ( 1 + 4k ^ {2} ) \frac{u ^ {4} }{4! } - ( 1 + 44k ^ {2} + 16k ^ {4} ) \frac{u ^ {6} }{6! } + \dots ,

where k is the modulus of the elliptic function, 0 \leq k \leq 1 ; \upsilon = u/2 \omega , and 2 \omega = \pi \theta _ {3} ^ {2} ( 0) . For k = 0, 1 one has, respectively, \mathop{\rm cn} ( u , 0) = \cos u , \mathop{\rm cn} ( u , 1) = 1/ \cosh u .

References

[1] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 2 , Springer (1964) pp. Chapt. 3

Comments

More on the function \mathop{\rm cn} u , e.g. derivatives, evenness, behaviour on the real line, etc. can be found in [a1].

References

[a1] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Cosine amplitude. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cosine_amplitude&oldid=46531
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article