Math 6420 • PDE

The \(z\)-transform is defined on sequences in \(\ell^2\) by
\[
  U(z)=\sum_{-\infty}^{\infty}u_nz^n,
\]
\[
  u_n=\frac{1}{2\pi i}\int_C U(z)z^{-n-1}\,dz
\]
where the contour \(C\) is a closed contour enclosing the origin, and integration is
counterclockwise. If \(z\) is restricted to lie on the unit circle then
\(U(z)=U(e^{i\theta})=f(\theta)\).
\begin{enumerate}
\item[a)] Find the relationship between the \(z\)-transform and the Fourier series
representation of \(f(\theta)\).
\item[b)] Use the \(z\)-transform to solve the system of equations
\[
  \frac{du_n}{dt}=\frac{1}{2h}(u_{n+1}-u_{n-1})
\]
with \(u_n(0)=\delta_{n0}\).
\end{enumerate}