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Could any one tell me how and what is the value of $\int_{|z|=2}ze^{3\over z}$ where the contour is oriented in anti clockwise direction. Can I apply Cauchy integral formula here?

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What is the Laurent series of $ze^{\frac{3}{z}}$ in a punctured neighborhood of the origin? You know that:

(1) $e^z=\sum_{n=0}^{\infty} \frac{z^n}{n!}$

is the Taylor series for the exponential function and if we substitute $\frac{3}{z}$ in (1), then we obtain:

(2) $e^{\frac{3}{z}}=\sum_{n=-\infty}^{0} \frac{z^n}{3^n n!}$;

the Laurent series of $e^{\frac{3}{z}}$ in a punctured neighborhood of the origin.

Exercise 1 (easy): Use (2) to determine the Laurent series of $ze^{\frac{3}{z}}$ in a punctured neighborhood of the origin.

Exercise 2: Calculate the residue of $ze^{\frac{3}{z}}$ at $z=0$; it's the coefficient of $\frac{1}{z}$ in the Laurent series of $ze^{\frac{3}{z}}$.

Exercise 3: Apply the Residue theorem!

If you hover your cursor over the grey box directly below, then you'll see the answer that you should get by doing the above computations. (I strongly encourage you to do the calculations yourself and subsequently check your answer by hovering your cursor over the grey box directly below.)

$9\pi i$

I hope this helps!

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More precisely the result you want to use is the residue theorem. The theorem applies to essential singularities, as well (as is the case here). Therefore what you need to do is calculate the residue at the origin.

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