Treffer: Elliptic Jacobi function and their generalizations in the case of arbitrary periods

Generalization of well-known elliptic Jacobi functions \(\text{sn}(u;k)\), \(\text{cn}(u;k)\) and \(\text{dn}(u;k)\) is considered. New elliptic functions \(s=\text{sne}(u; \lambda,k)\) and \(c=\text{cne}(u; \lambda,k)\) are inverse functions to the following elliptic integrals \[ u=\int_0^s\frac{dt}{\sqrt{(1-\lambda^2t^2)(1-k^2t^2)}},\quad u=\int_c^1\frac{dt}{\sqrt{(1-t^2)(\lambda^2-k^2+k^2t^2)}}. \] Relations for the functions \(s=\text{sne}(u;\lambda,k)\), \(c=\text{cne}(u;\lambda,k)\) and \(\delta=\text{dne }(u;\lambda,k)\) that are similar to those valid for Jacobi functions \(\text{sn}(u;k)\), \(\text{cn}(u;k)\) and \(\text{dn}(u;k)\) are obtained. For example, expression of \(\text{sne}(u+v;\lambda,k)\) through the functions \(\text{sne }(u;\lambda,k)\), \(\text{cne}(u; \lambda,k)\) and \(\text{ dne}(u;\lambda,k)\) is given. Poles, zeros and periods of generalized Jacobi functions are calculated too.