I am having trouble understanding a step in the proof of Lemma 1 of "A New Constructive Proof of the Malgrange-Ehrenpreis Theorem" by Wagner, 2009. It should be a simple result involving Cauchy's theorem and the residue theorem, but its been a while since I've looked at this and I could do with a hand. The particular step I am having trouble with is the following:
Given the pairwise different complex numbers $\lambda_0, ..., \lambda_m\in\mathbb{C}$, define the polynomial $p(z) = \prod\limits_{j=0}^m (z-\lambda_j)$. Then \begin{equation} \frac{1}{2\pi i}\lim_{N\to\infty} \int\limits_{|z|=N} \frac{z^k}{p(z)}dz = \begin{cases} 0 & \text{if $k = 0,...,m-1$},\\ 1 & \text{if $k = m$}. \end{cases} \end{equation}
I can understand that the limit goes to 0 for $k<m$ but I don't see how it is 1 when $k=m$.
Thanks in advance!
Notice that the degree of $p$ is $m+1$. So when $k = m$, the integrand is $$ \frac{z^m}{z^{m+1} + \text{lower order terms}} \\ = \frac{1}{z} \cdot \frac{1}{1 + \text{terms with negative powers of $z$}} \text{.} $$ In the limit as $|z| = N \rightarrow \infty,$ the terms with negative powers of $z$ go to $0$, so the integrand approaches $1/z$. Hopefully, you know the residue of $1/z$.
If we wanted to be careful, we would estimate the maximal discrepancy between $1$ and $\frac{1}{1 + \text{terms with negative powers of $z$}}$ on the circle of radius $N$. We should be able to show that the discrepancy is upper bounded by $k/N$ for some constant $k$ depending on the coefficients of $p$. Do we need that level of care?