SUMMARY
The integral \(\int_{|z-2i|=2} \frac{dz}{z^2-9}\) evaluates to zero because the singularities at \(z=3\) and \(z=-3\) lie outside the contour defined by \(|z-2i|=2\). Since the integrand \(\frac{1}{z^2-9}\) is analytic within the region enclosed by the contour, the Cauchy Integral Theorem confirms that the integral over a closed contour of an analytic function is zero. Thus, the conclusion is definitive: the integral evaluates to zero.
PREREQUISITES
- Complex analysis fundamentals
- Cauchy Integral Theorem
- Understanding of analytic functions
- Knowledge of contour integration
NEXT STEPS
- Study the Cauchy Integral Theorem in detail
- Learn about singularities and their impact on contour integrals
- Explore examples of contour integration with different contours
- Investigate the properties of analytic functions in complex analysis
USEFUL FOR
Students of complex analysis, mathematicians focusing on contour integration, and anyone seeking to understand the implications of singularities in integrals.