The addition theorem of Weierstrass’s sigma function. (English) Zbl 0791.39009
Summary: Weierstraß’s \(\sigma\)-function is a solution \(\theta\) of the following functional equation (1) \(\theta(u+v_ 1)\theta (u-v_ 1) \theta(v_ 2+v_ 3) \theta(v_ 2-v_ 3)+\theta (u+v_ 2) \theta (u- v_ 2) \theta (v_ 3+v_ 1) \theta(v_ 3-v_ 1)+\theta (u+v_ 3) \theta (u-v_ 3) \theta (v_ 1+v_ 2) \theta (v_ 1-v_ 2)=0\). This is the so-called addition theorem for the \(\sigma\)-function.
In this paper all continuous solutions \(\theta:\mathbb{R}^ n \to \mathbb{C}\) of (1) for arbitrary \(n \in \mathbb{N}\) are found. It turns out that the \(\sigma\)-function is in essence the only solution of the functional equation (1). This result is obtained from investigations which are centered around the functional equation \(\chi (u+v) \varphi (u-v)=\sum^ k_{\nu=1} f_ \nu (u) g_ \nu(v)\), which is well-known in connection with the addition theorems for theta functions.
In this paper all continuous solutions \(\theta:\mathbb{R}^ n \to \mathbb{C}\) of (1) for arbitrary \(n \in \mathbb{N}\) are found. It turns out that the \(\sigma\)-function is in essence the only solution of the functional equation (1). This result is obtained from investigations which are centered around the functional equation \(\chi (u+v) \varphi (u-v)=\sum^ k_{\nu=1} f_ \nu (u) g_ \nu(v)\), which is well-known in connection with the addition theorems for theta functions.
MSC:
| 39B32 | Functional equations for complex functions |
| 33E05 | Elliptic functions and integrals |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Keywords:
Weierstrass’s sigma function; addition theorem; continuous solutions; functional equation; theta functionsReferences:
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