Evaluation of Complex Integral Using Cauchy's Integral Formula: A Detailed Guide

Understanding how to evaluate complex integrals using Cauchy's Integral Formula is crucial in complex analysis. This article will guide you through the process of evaluating the integral of (frac{z^2 - 4z 4}{z - i}) over a specific contour. We will focus on the application of Cauchy's integral formula and the steps involved in solving such problems.

Introduction to Cauchy's Integral Formula

Cauchy's Integral Formula is a cornerstone in complex analysis, providing a straightforward method to evaluate certain types of integrals involving analytic functions. It states that for a function (f(z)) that is analytic inside and on a simple closed contour (C), and for any point (a) inside (C), the integral of (f(z)) over (C) is given by:

[f(a) frac{1}{2pi i} oint_{C} frac{f(z)}{z - a} dz]

Evaluating the Given Contour Integral

Consider the integral we are tasked with evaluating:

(I oint_{z2} frac{z^2 - 4z 4}{z - i} dz)

The numerator, (z^2 - 4z 4), is a polynomial, making it an entire function (analytic everywhere). The only singularity of the integrand occurs where the denominator is zero, i.e., (z - i 0), which gives (z i).

Since (z i) is the only singularity and it lies inside the closed contour (z 2), we can apply Cauchy's Integral Formula:

(I 2pi i cdot f(i))

where (f(z) z^2 - 4z 4).

Step-by-Step Solution Using Cauchy's Integral Formula

1. Identify the function (f(z)) and the singularity inside the contour.

2. Apply Cauchy's Integral Formula:

(I 2pi i cdot f(i))

3. Substitute (i) into (f(z)) to find (f(i)):

(f(i) i^2 - 4i 4)

4. Calculate the value of (f(i)):

(f(i) -1 - 4i 4 3 - 4i)

5. Substitute back into Cauchy's Integral Formula:

(I 2pi i cdot (3 - 4i))

6. Simplify the expression:

(I 2pi i cdot 3 - 2pi i cdot 4i 6pi i - 8pi)

Conclusion

The integral (I oint_{z2} frac{z^2 - 4z 4}{z - i} dz) evaluates to (-8 6pi i). This approach exemplifies the power and utility of Cauchy's Integral Formula in solving problems in complex analysis.

Keywords:

Cauchy's Integral Formula Contour Integration Complex Analysis

References:

Primer on complex analysis by Lars V. Ahlfors, 1953. Complex Variables and Applications by James Ward Brown and Ruel V. Churchill, 1996.