### A Note on the Elliptic Integral K(k).

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The aim of this paper is to study a generalized form of elliptic-type integrals which unify and extend various families of elliptic-type integrals studied recently by several authors. In a recent communication [1] we have obtained recurrence relations and asymptotic formula for this generalized elliptic-type integral. Here we shall obtain some more results which are single and multiple integral formulae, differentiation formula, fractional integral and approximations for this class of generalized...

We give a geometric descriptions of (wave) fronts in wave propagation processes. Concrete form of defining function of wave front issued from initial algebraic variety is obtained by the aid of Gauss-Manin systems associated with certain complete intersection singularities. In the case of propagations on the plane, we get restrictions on types of possible cusps that can appear on the wave front.

A new connection between geometric function theory and number theory is derived from Ramanujan’s work on modular equations. This connection involves the function ${\phi}_{K}\left(r\right)$ recurrent in the theory of plane quasiconformal maps. Ramanujan’s modular identities yield numerous new functional identities for ${\phi}_{1/p}\left(r\right)$ for various primes p.