The Values of the Weierstrass Phe-function at the Zeros of its Derivative are nonvanishing

An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/

Goals: * To find the zeros of the derivative of the Weierstrass phe-function associated to a lattice

* To use the ODE established in the previous lecture to analyze the values of the Weierstrass phe-function at the zeros of its derivative and to show that these values are nonvanishing analytic (holomorphic) functions on the upper half-plane

* To introduce the notion of order for an elliptic function, namely the finite positive integer which is the number of times the function assumes any value in the extended complex plane (Riemann sphere)


Keywords: Upper half-plane, invariants for complex tori, lattice (or) grid in the plane, fundamental parallelogram (or) period parallelogram associated to a lattice, complex torus associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, simple zero, pole of order three, isolated double pole, Argument Principle, Residue theorem, order of an elliptic function, automorphic function (or) automorphic form, modular function (or) modular form, congruence mod two subgroup of the unimodular group, even function, odd function