<s>[INST] Computing $\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz$ using Cauchy integral formula Let $\alpha(t) = re^{it}$ where $|a|<r<|b|$ and $t \in [0,2\pi]$. I'd like to compute
$$\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz \ \ \ \ n, m \in \mathbb{N}.$$
It appears that the answer is
$$2\pi i (-1)^m{n + m -2 \choose n-1}\frac{1}{(b-a)^{n+m-1}}.$$
I try to compute it but I'm not sure how that's the final answer. Here is my attempt:
Let $f(z) = \frac{1}{(z-b)^m}$, then by Cauchy integral formula
$$\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz = \int_{\alpha}\frac{f(z)}{(z-a)^n}dz = \frac{2\pi i}{(n-1)!}f^{(n-1)}(a).$$
Trying to find an expression of $f^{(n-1)}(z)$
\begin{align*}
f^{(1)} &= -m\frac{1}{(z-b)^{m-1}}\\
f^{(2)} &= m(m-1)\frac{1}{(z-b)^{m-2}}\\
f^{(3)} &= -m(m-1)(m-2)\frac{1}{(z-b)^{m-3}}\\
&\vdots\\
f^{(n-1)} &= (-1)^{n-1}m(m-1)(m-2)\dotsc(m-n + 2)\frac{1}{(a-b)^{m-n+1}}
\end{align*}
I think this should be correct. So,
\begin{align*}
\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz &= \frac{2\pi i}{(n-1)!}(-1)^{n-1}m(m-1)(m-2)\dotsc(m-n + 2)\frac{1}{(a-b)^{m-n+1}}\\
&= \frac{2\pi i}{(n-1)!}(-1)^{2n-2-m}m(m-1)(m-2)\dotsc(m-n + 2)\frac{1}{(b-a)^{m-n+1}}\\
&=\frac{2\pi i}{(n-1)!}(-1)^{m}m(m-1)(m-2)\dotsc(m-n + 2)\frac{1}{(b-a)^{m-n+1}}.
\end{align*}
But the exponent on $(b-a)$ is not the same. Further, I have no clue how I can get the binomial coefficient. Writing out the binomial coefficient in the given answer doesn't seem to help me either.
$${n + m -2 \choose n-1} = \frac{(n+m-2)!}{(n-1)!(m-1)!} = \frac{(n+m-2)(n+m-3)\dotsc 2\cdot 1}{(n-1)!(m-1)!} $$
NOTE: I haven't learned residue theorem yet.
 [/INST] Answering my own question, thanks to the comments.
Let $f(z) = \frac{1}{(z-b)^m}$, then by Cauchy integral formula
$$\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz = \int_{\alpha}\frac{f(z)}{(z-a)^n}dz = \frac{2\pi i}{(n-1)!}f^{(n-1)}(a).$$
Trying to find an expression of $f^{(n-1)}(z)$
\begin{align*}
f^{(1)} &= -m\frac{1}{(z-b)^{m+1}}\\
f^{(2)} &= m(m+1)\frac{1}{(z-b)^{m+2}}\\
f^{(3)} &= -m(m+1)(m+2)\frac{1}{(z-b)^{m+3}}\\
&\vdots\\
f^{(n-1)} &= (-1)^{n-1}m(m+1)(m+2)\dotsc(m+n - 2)\frac{1}{(a-b)^{m+n-1}}
\end{align*}
So,
\begin{align*}
\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz &= \frac{2\pi i}{(n-1)!}(-1)^{n-1}m(m+1)(m+2)\dotsc(m+n - 2)\frac{1}{(a-b)^{m+n-1}}\\
&= \frac{2\pi i}{(n-1)!}(-1)^{n-1-m-n+1}m(m+1)(m+2)\dotsc(m+n - 2)\frac{1}{(b-a)^{m+n-1}}\\
&=\frac{2\pi i}{(n-1)!}(-1)^{-m}m(m+1)(m+2)\dotsc(m+n - 2)\frac{1}{(b-a)^{m+n-1}}\\
&=\frac{2\pi i}{(n-1)!}(-1)^{m}m(m+1)(m+2)\dotsc(m+n - 2)\frac{1}{(b-a)^{m+n-1}}.
\end{align*}
Note that
$${n + m -2 \choose n-1} = \frac{(n+m-2)(n+m-3)\dotsc m}{(n-1)!}.$$
Hence,
$$\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz = 2\pi i(-1)^m{n + m -2 \choose n-1}\frac{1}{(b-a)^{m+n-1}}.$$
</s>