Uniform asymptotics of the sine amplitude function

Two cases of degeneration of elliptic functions are well known: degeneration into trigonometric functions and degeneration into hyperbolic functions. Approximations of an elliptic function in a neighborhood of a degeneration are usually examined by means of series in the modulus of the elliptic function. For applications of the theory of elliptic functions in the theory of dynamical systems, uniform approximations with respect to the modulus and the independent variable are important. This review contains methods for constructing uniform asymptotics.