# Sine amplitude

elliptic sine

One of the three basic Jacobi elliptic functions, written as

$$\mathop{\rm sn} u = \mathop{\rm sn} ( u, k) = \sin \mathop{\rm am} u .$$

The sine amplitude can be defined by theta-functions or by means of a series in the following way:

$$\mathop{\rm sn} u = \mathop{\rm sn} ( u, k) = \ \frac{\theta _ {3} ( 0) }{\theta _ {2} ^ \prime ( 0) } \frac{\theta _ {1} ( v) }{\theta _ {0} ( v) } =$$

$$= \ u - ( 1 + k ^ {2} ) \frac{u ^ {3} }{3! } + ( 1 + 14k ^ {2} + k ^ {4} ) \frac{u ^ {5} }{5! } - \dots ,$$

where $k$ is the modulus of the sine amplitude (usually $0 \leq k \leq 1$) and $v = u/2 \omega$, $2 \omega = \pi \theta _ {3} ^ {2} ( 0)$. When $k = 0, 1$, respectively, $\mathop{\rm sn} ( u, 0) = \sin u$, $\mathop{\rm sn} ( u, 1) = \mathop{\rm tanh} u$.

#### References

 [1] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 2 , Springer (1964) pp. Chapt. 3
How to Cite This Entry:
Sine amplitude. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sine_amplitude&oldid=48716
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article