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