Complex Contour Integration and Exponential Functions

Here is a classic problem from complex analysis. We need to evaluate the contour integral of e to the z over z squared minus four.

The first step is always to identify the singularities of the integrand. We do this by setting the denominator to zero.

We can factor the denominator z squared minus four into z minus two times z plus two.

This tells us we have simple poles at z equals two and z equals negative two.

Now, the value of the integral depends entirely on the contour C. The problem image cuts off the definition, but a standard contour for this problem is the circle modulus of z minus one equals two.

Let's check which poles lie inside this contour. The center is at z equals 1, and the radius is 2.

For the pole at 2: the distance from the center 1 is just 1 unit. Since 1 is less than the radius 2, this pole is INSIDE.

For the pole at negative 2: the distance from the center 1 is 3 units. Since 3 is greater than the radius 2, this pole is OUTSIDE.