Su una equazione funzionale proveniente dalla teoria delle funzioni ellittiche jacobiane. (On a functional equation originating from the theory of Jacobi elliptic functions). (Italian. English summary) Zbl 0666.39004
A proof is offered that the following are the only solutions of \(g(z_ 1+z_ 2)-g(z_ 1)-g(z_ 2)=f(z_ 1)\cdot f(z_ 2)\cdot f(z_ 1+f_ 2),\) analytic on a (complex) neighbourhood of the origin: \(f(z)=a\), \(g(z)=cz-a^ 3\); \(f(z)=az\), \(g(z)=cz+a^ 3z^ 3/3\) and \(f(z)=a sn(bz,k)\), \(g(z)=cz+a^ 3\int^{bz}_{0}sn^ 2(t,k)dt\) (a, b, c, k are constants; sn the elliptic sine).
Reviewer: J.Aczél
MSC:
| 39B62 | Functional inequalities, including subadditivity, convexity, etc. |
| 33E05 | Elliptic functions and integrals |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
| 34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
