SUMMARY
The discussion centers on evaluating the integral \( \oint\frac{e^{iz}}{z^3}dz \) over a square contour centered at 0 with sides greater than 1. The integral contains a pole of order 3 at \( z=0 \). Utilizing the Cauchy Integral Formula (CIF), the result is calculated as \( -\pi i \) by applying the formula \( \oint\frac{f(z)}{(z-z_0)^{n+1}}dz = \frac{2\pi i}{n!} f^{(n)}(z_0) \) with \( n=2 \). Verification through Laurent series expansion confirms the coefficient of \( z^{-1} \) aligns with the computed result.
PREREQUISITES
- Understanding of complex analysis, specifically contour integration
- Familiarity with the Cauchy Integral Formula
- Knowledge of Laurent series and residue theory
- Ability to differentiate complex functions
NEXT STEPS
- Study the application of the Cauchy Integral Formula in various contexts
- Explore the properties of Laurent series and their coefficients
- Learn about higher-order poles and their residues in complex analysis
- Investigate contour integration techniques for different shapes and functions
USEFUL FOR
Students and professionals in mathematics, particularly those specializing in complex analysis, as well as educators looking to deepen their understanding of contour integration techniques.