Let , $f:\Bbb C \to \Bbb C$ be an entire function and $\displaystyle \int_{\Bbb R^2}|f(x+iy)|\,dx\,dy<\infty$. Then prove that $f(z)=0$ for all $z\in \Bbb C$.
Put $x=r\cos \theta$ and $y=r\sin \theta$. Then , integration becomes $\displaystyle \int_{r=0}^{\infty}\int_{\theta=0}^{2\pi}|f(re^{i\theta})|r\,d\theta\,dr<\infty$. But from here I'm unable to conclude nothing.! How I proceed? Any hint please.
Hint: Use the estimate from Cauchy's integral formula $$ \lvert f^{(n)}(0) \rvert \le \frac{n!}{2 \pi r^n} \int_0^{2\pi} \lvert f(re^{i\theta}) \rvert \, d\theta $$ to show that $$ \int_0^{\infty}\int_0^{2\pi}|f(re^{i\theta})|r\,d\theta \, dr $$ cannot be finite unless $f^{(n)}(0) = 0$ for all $n \ge 0$.