Consider the Fourier transform in complex background. $z\in\mathbb{C}$, $F(z)=\int_{-\infty}^{\infty}f(x)e^{-izx}\mathrm{d}x$ is the complex Fourier transform of real-valued function $f(x)$. The function takes zero value when $|x|>M$. I proved the transform defines a complex function and is holomorphic.
If $F(z)=0$ for $|z|>M$, I want to know the value of $f(x)$ at point $x=0$, so I tried to apply the inversion formula, $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(z)e^{izx}\mathrm{d}z$$ I plugged $x=0$ in, found $f(0)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(z)\mathrm{d}z$, that is $$f(0)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}f(x)e^{-izx}\mathrm{d}x\right)\mathrm{d}z$$ but it seems no use. Or by condition $F(z)=0,|z|>M$, I again tried $$f(0)=\frac{1}{2\pi}\int_{?}^{?}F(z)\mathrm{d}z$$ when the module of complex number $z$ is not larger than $M$.