Is $\displaystyle{\exp ^{z^2}}$ periodic?
I was trying to find about the periodicity of any function of the form $\displaystyle{\exp ^{p(z)}}$, where $p$ is a complex polynomial. Since $\exp z$ is periodic with period $2\pi i$, I have tried it for $\displaystyle{\exp ^{z^2}}$ and have found that it's period is not $2\pi i$. I wanted to know whether is periodic in the first place. Please help. Thanks.
$$e^{(z+t)^2}=e^{z^2}e^{2zt}e^{t^2}=e^{z^2}$$ requires $$2zt+t^2=i2k\pi.$$
The nonzero solutions for $t$ do depend on $z$, there is no period.
The conclusion will be the same for any nonlinear polynomial.