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Q:

Calculating a complex integral using Cauchy’s integral formula

Here is the problem:
$$f(z)=e^z\,\frac{e^{z^2}-1}{z(z^2+1)}$$
is an entire function, which satisfies the Cauchy’s integral formula:
$$f^{(n)}(0)=\frac{n!}{2\pi i}\int_{|z|=r}\frac{f(z)}{z^{n+1}}dz,\quad n\ge0$$
Then the residue of $f(z)$ at $z=0$ is:
$$\text{Res}(f,0)=\lim_{z\to0}\frac{z^n}{n!}f^{(n)}(z)=\lim_{z\to0}\frac{e^{z^2}-1}{z(z^2+1)z^n}=\frac{e-1}{(1+\frac{1}{4})}$$
Why do we obtain this result? I am working on complex analysis and I have a hard time implementing the Cauchy’s integral formula. If you could provide step by step details I would appreciate.
Thank you!

A:

To calculate the limit, we write the expression as a power series for $|z|

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