SUMMARY
The function \(\frac{e^{ikz}}{z^{2}+m^{2}}\) possesses two simple poles located at \(im\) and \(-im\). However, when \(k > 0\), the contour integral is evaluated using a closed contour in the upper half-plane, which only encloses the pole at \(im\). Consequently, only the pole at \(im\) contributes to the residue in this scenario, leading to the conclusion that the integral effectively considers just one pole for positive \(k\).
PREREQUISITES
- Complex analysis fundamentals
- Understanding of contour integration
- Knowledge of residues and poles
- Familiarity with the exponential function in complex variables
NEXT STEPS
- Study the residue theorem in complex analysis
- Learn about contour integration techniques in the upper half-plane
- Explore the implications of pole contributions in integrals
- Investigate the behavior of functions with multiple poles
USEFUL FOR
Mathematicians, physics students, and anyone studying complex analysis or evaluating integrals involving complex functions.