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

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