Providing background from the previous parts of the question, we are asked:
Define: $$\hat f(z)=\int_{-\infty}^{\infty}f(t)e^{-2\pi izt}dt$$ where $f$ is entire and is smooth ($\mathbf{C^2}$). Let: $$P(z)=a_n(2\pi iz)^n+...+a_0$$ where $a_j$ is a complex constant. Set $u(t)=\int_L\frac{e^{2\pi izt}}{P(z)}\hat f(z)dz$, where $L$ is the line along which $P(z)$ does not vanish. Check that: $$\sum_{j=0}^na_j\Big(\frac{d}{dt}\Big)^ju(t)=\int_Le^{2\pi izt}\hat f(z)dz$$ and: $$\int_Le^{2\pi izt}\hat f(z)dz=\int_{-\infty}^{\infty}e^{2\pi ixt}\hat f(x)dx$$ Conclude by Fourier Inversion Theorem that: $$\sum_{j=0}^na_j\Big(\frac{d}{dt}\Big)^ju(t)=f(t)$$
While I understand the intuition behind the problem, I'm slightly confused as to how to proceed. I understand that it's a fairly straightforward calculation, but for one reason or another I'm not getting the desired result. Any suggestions would be greatly appreciated.