An addition formula for the Jacobian theta function and its applications

作者：Liu Zhi Guo^*

来源：Advances in Mathematics, 2007, 212(1): 389-406.

DOI：10.1016/j.aim.2006.10.005

• Weierstrass sigma function
• addition formula
• elliptic functions
• Jacobi's theta function
• Winquist's identity
• Ramanujan's cubic theta function theory

In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary.

• 出版日期2007-6-20
• 单位华东师范大学