a) Let $\displaystyle \widehat{f}(z)=\int_{-\infty}^{\infty} f(t) e^{-2 \pi i zt} dt$

Assume that $\displaystyle f$ has bounded support and smooth.

Show, by integration by parts that $\displaystyle \mid \widehat{f}(x+iy) \mid \leq \frac{A}{1+x^2}$ if $\displaystyle \mid y \mid \geq 0$

b) Write $\displaystyle P(z)=a_n(2\pi iz)^n + a_{n-1}(2\pi iz)^{n-1} +...+a_0$ where $\displaystyle a_i$ are complex constants.

Find a real number $\displaystyle c$ such that $\displaystyle P(z)$ does not vanish on the line $\displaystyle L= \{z:z=x+ic, x\in R \}$

For a) I let $\displaystyle u=f(t)$ and $\displaystyle dv=e^{-2 \pi i zt}dt$. Then integrate by parts. Is this the right approach?