An Important Theta Function Identity And Its Applications

The study of Theta Function is a very hot field in Special Function,Theta function is a holomorphic function with double-periods,and the elliptic function is a merornorphic function of double-periods.As a result of theta function and elliptic function of this feature, we usually use theta functions to construct rational characteristics elliptic functions to study some the mathematical problem.In this paper we use theta functions to construct an rational characteristics elliptic function, derive an important theta function identity.then we discuss Theta functions with f(z+π)=-f(z)和f(z+πτ)=-q^-5e^-10izf(z) .Specific operation is as follows: According the doubly-periodic Characteristic of Elliptic Function and the sum of all the residues of an elliptic function in the period parallelogram is zero, we use the residue theorem of elliptic functions,give the main theorem of this paper.[see 2.1] Then we choose theta functions that satisfy the above characters, and take them into the main theorem, when we choose properly Parameters, we derive Jacobi Theta Identity and some other Jacobi and Ramanujan modular equations. We also derive some new identities with this method.