A nice way to remember a formula is to connect it to something you already know. For example, to remember the Cauchy integral formula, I remember that $f(z_0)=\frac{1}{2\pi i}\int_\Gamma\frac{f(z)}{z-z_0}\,dz$ because if you write out the Laurent series for $\frac{f(z)}{z-z_0}$ you're likely to get something like:
$$ \frac{f(z)}{z-z_0}=\frac{a_{-1}}{z-z_0}+a_0 +a_1(z-z_0)+\dots \,\,\,\,(1)$$
If you take the integral of the right hand side of (1), you're going to be left with $a_{-1} 2\pi i$, but we were really looking for the behavior of $f(z)$ at $z_0$, and since we used an integral to find the behavior we got $2\pi i$ as an artifact, so we're just going to divide our answer by $2\pi i$ and be left with $f(z_0)$ a.k.a $Res(f;z_o).$
My questions:
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is there a similar way to think about $f^{(n)}(z_0)=\frac{n!}{2\pi i}\int_\Gamma\frac{f(z)}{(z-z_0)^{n+1}}\,dz$? I've tried writing out a Laurent series for $\frac{f(z)}{(z-z_0)^{n+1}}\,dz$, but it didn't get me anywhere.
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if $f(z)=\frac{g(z)}{(z-z_0)^m}$, is $g(z_0)=a_{-1}$ ? (where $a_{-1}$ is from its Laurent series)
Obviously there is a strong connection between the singularities of a function, its Laurent series, residue theory and the Cauchy integral formulas. After a semester of introductory complex analysis, I'm struggling to place everything in a coherent narrative. Any help would be greatly appreciated.
Edit: Based on the answers I've formed a new narrative to remember the formulas. I thought I'd share it in case someone finds it helpful.
Let's say we have a function:
$$ \frac{ f(z) }{ (z-z_0)^{n+1} } $$
We want to study the behavior of $f^{(n)}(z)$ at $z=z_0$. We know that a Taylor coefficient is on the form $a_n=\frac{f^{(n)}(z_0)}{n!}$, so we can find $f^{(n)}(z_0)$ by manipulating the Taylor series.
So expand to Taylor form:
$$ \frac{ f(z) }{ (z-z_0)^{n+1} }=\frac{c_0}{(z-z_0)^{n+1}}+\frac{c_{1}}{(z-z_0)^{n}} +\dots+\frac{c_{k}}{(z-z_0)^{1}}+c_{k+1}+c_{k+2}(z-z_0)+\dots $$
We see that we can recover $c_k=\frac{f^{(k)}(z_0)}{n!}$ by integrating the equation above. But since we used an integral to isolate $c_k$, we get $2\pi i$ as an artifact, and since we used the Taylor series we also got $\frac{1}{n!}$ as an artifact.
We can think of the forumla $f^{(n)}(z_0)=\frac{n!}{2\pi i}\int_\Gamma\frac{f(z)}{(z-z_0)^{n+1}}$ a formula for recovering $f^{(n)}(z_0)$ from a Taylor series: Use $\int_\Gamma\frac{f(z)}{(z-z_0)^{(n+1)}}$ to isolate the Taylor coefficient and remove the extraneous information by multiplying by $\frac{n!}{2\pi i}$.
When $n=0$ it's just the regular Cauchy integral formula, of course; $\frac{0!}{2\pi i}=\frac{1}{2\pi i}$.
Best Answer
Expand $f$ into its Taylor series,
$$f(z) = \sum_{k=0}^\infty c_k(z-z_0)^k.$$
Then you have
$$\frac{f(z)}{(z-z_0)^{n+1}} = \sum_{k=0}^\infty c_k (z-z_0)^{k-n-1}.$$
When you integrate termwise, the only term with a nonzero integral is the one with the exponent $k-n-1 = -1$, that is, $k = n$, so the integral is $2\pi i c_n$. But $$c_k = \frac{f^{(k)}(z_0)}{k!}$$ for all $k$.