Cosine amplitude

elliptic cosine

One of the three basic Jacobi elliptic functions, denoted by

$$ \mathop{\rm cn} u = \ \mathop{\rm cn} ( u , k) = \ \cosam u . $$

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)