Say that a function on the complex plane $\mathbb{C}$ is periodic if $f(z+1)=f(z)$ and $f(z+i)=f(z)$ for all $z$. If $f$ is a periodic analytic function, show that $f$ is constant.

If $f$ is a meromorphic periodic function, show that the number of zeros of $f$ in the square $[0,1) \times[0,1)$ is equal to the number of poles, both counted with multiplicities.

Define

$$f(z)=\frac{1}{z^{2}}+\sum_{w}\left[\frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right]$$

where the sum runs over all $w=a+b i$ with $a$ and $b$ integers, not both 0 . Show that this series converges to a meromorphic periodic function on the complex plane.