Residue Calc: $\cot^n(z)$ at $z=0$

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SUMMARY

The residue of the function $\cot^{n}(z)$ at the point $z=0$ is definitively established as $\sin \left( \frac{n \pi}{2}\right)$ for all natural numbers $n$. This conclusion is derived from the properties of the cotangent function and its behavior around singularities. The analysis confirms that the residue varies based on the parity of $n$, with specific values for even and odd integers.

PREREQUISITES
  • Understanding of complex analysis, specifically residue theory.
  • Familiarity with the cotangent function and its Taylor series expansion.
  • Knowledge of trigonometric identities and their applications in complex functions.
  • Basic skills in evaluating limits and singularities in complex functions.
NEXT STEPS
  • Study the properties of residues in complex analysis.
  • Explore the Taylor series expansion of $\cot(z)$ around $z=0$.
  • Investigate the implications of residue calculations in contour integration.
  • Learn about the relationship between residues and the evaluation of integrals in complex analysis.
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in advanced calculus and residue theory will benefit from this discussion.

polygamma
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Show that the residue of $\cot^{n}(z)$ at $z=0$ is $\sin \left( \frac{n \pi}{2}\right)$, $n \in \mathbb{N}$.
 
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Hint:

Integrate $\cot^{n} (z)$ around a rectangular contour with vertices at $- \frac{\pi}{2} - iR$, $ \frac{\pi}{2} - i R$, $\frac{\pi}{2} + iR$, and $- \frac{\pi}{2} + iR$. Then let $ R \to \infty$.
 

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