A contour integral !

[Tolaso J Kos]

Let \( \displaystyle f(z)=\frac{1}{z}.\frac{1-2z}{z-2}\cdots \frac{1-10z}{z-10} \). Evaluate the counter clockwise contour integral:

$$\oint \limits_{|z|=100}f(z)\,{\rm d}z$$

[galactus]

I used the residue at infinity and changed it to 0 by using

$$Res(f(z), \infty)=-Res\left(\frac{1}{z^{2}}f(1/z), \;\ 0\right)$$

$$-2\pi i \cdot \lim_{z\to 0}\frac{1}{z}\cdot \frac{\prod_{k=0}^{5}(1-2k/z)}{\prod_{k=0}^{5}(1/z-2k)}$$

$$=-2\pi i \cdot 3840$$


One could also add up the residues of the poles from 0 to 10 using the even digits and arrive at the same answer.