Evaluating Complex Line Integrals Using the Residue Theorem

Complex analysis, a branch of mathematical analysis, deals with the study of functions of complex numbers. One of the key tools in this field is the Residue Theorem, which is particularly useful for evaluating complex line integrals such as the one we will examine in this article.

Understanding the Problem

We are tasked with evaluating the integral ∫C (3z2 - 7z - 1)/(z - 1) dz, where C is the circle |z| 1.5. To do this, we will employ the powerful Residue Theorem from complex analysis.

Identifying Singularities

The first step in applying the Residue Theorem is to identify the singularities of the integrand. The given integrand 3z2 - 7z - 1/z - 1 has a singularity at z -1. Since the contour C has a radius of 1.5, the singularity located at z -1 lies inside the contour.

Calculating the Residue

To find the residue of the integrand at this simple pole, we need to rewrite the integrand around the singularity. This can be done by performing a polynomial long division:

Divide 3z2 by z to get 3z. Multiply 3z by z - 1 to get 3z2 - 3z. Subtract this from 3z2 - 7z - 1 to get 4z - 1. Divide 4z by z to get 4. Multiply 4 by z - 1 to get 4z - 4. Subtract this from 4z - 1 to get -3.

Thus, we can rewrite the integrand as:

3z - 4 - 3/(z - 1)

The residue at the pole z -1 is the coefficient of 1/z - 1, which is -3.

Applying the Residue Theorem

Now, we apply the Residue Theorem:

∫C fz dz 2πi · (sum of residues inside C)

Since there is only one residue at -1, we have:

∫C (3z2 - 7z - 1)/(z - 1) dz 2πi · (-3) -6πi

Therefore, the value of the integral is boxed(-6πi).

Cauchy's Integral Formula Approach

We can also solve this problem using Cauchy's Integral Formula. According to Cauchy's Integral Formula, for a function f(z) that is analytic inside and on a closed contour C and for a point z a inside C, we have:

∫C f(z)/(z - a) dz 2πi · f(a)

In this case, f(z) 3z2 - 7z - 1 and a -1. Plugging these into the formula, we get:

∫C (3z2 - 7z - 1)/(z - 1) dz 2πi · (3(-1)2 - 7(-1) - 1) 2πi · (3 7 - 1) -6πi

Alternate Calculation Using Cauchy's Theorem

Alternatively, since the only singularity, z -1, does not lie inside the circle |z| 1.5, the integrand is analytic inside and on the contour C. By Cauchy's Theorem, the integral of an analytic function over a closed contour is zero:

∫C (3z2 - 7z - 1)/(z - 1) dz 0

Conclusion

In conclusion, by either using the Residue Theorem or Cauchy's Integral Formula, we have demonstrated that the value of the integral is -6πi. Understanding and applying these theorems is crucial for solving complex problems in complex analysis.