# Help me solving this complex integral

• sabbagh80
In summary, the integral provided is over the unit circle with two poles at z=0 and z=1 and the main problem is the term exp(b/z). To solve it, the Laurent series for e^z and e^(1/z) must be computed and the coefficients for the 1/z term must be picked out. The final result is pi minus 2pi times the sum of the coefficients for the two poles.
sabbagh80
Hi,

$\oint \frac{e^{-(a+b)+az+\frac{b}{z}}}{z(z-1)}dz$

over the unit circle, where $a, b$ are two positive constants (it is not a homework)

There are two poles, one in the centre (no problem) and on on the boundary (not that much of a problem, you just have to deform your contour slightly. I think you main problem is going to be exp(b/z) term.

hunt_mat said:
There are two poles, one in the centre (no problem) and on on the boundary (not that much of a problem, you just have to deform your contour slightly. I think you main problem is going to be exp(b/z) term.

I think the problem is exactly related to the pole which is placed at $z=0$. it is of order infinity. Am I right?

So the integral basically becomes:
$$e^{-(a+b)}\oint_{\gamma}\frac{e^{az+\frac{b}{z}}}{z(z-1)}dz$$
According to the sources I have read, you have to compute the Laurent series for $e^{z}$ and te Laurent series of $e^{\frac{1}{z}}$ along with all the other functions involved and just pick out the coefficient of the $1/z$ term. Sorry, but it is going to take a lot of algebra on this one.

Residue at pole z=1 is $2\pi \frac{1}{2}$
and residue at pole z=0 is $-2\pi e^{-(a+b)}\sum _{n=0}^{\infty} \frac{a^n}{n!} \sum_{m=0}^{n}\frac{b^m}{m!}$

So, we conclude the result as:

$\pi - 2\pi e^{-(a+b)}\sum _{n=0}^{\infty} \frac{a^n}{n!} \sum_{m=0}^{n}\frac{b^m}{m!}$

Is everything Ok?

## 1. How do I approach solving a complex integral?

When faced with a complex integral, it is important to first understand the basic concepts and techniques of integration. This includes being familiar with different integration methods such as u-substitution, integration by parts, and trigonometric substitution. It is also helpful to have a good understanding of the properties of integrals and how they can be used to simplify complex expressions.

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## 4. How do I know if I have solved a complex integral correctly?

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