- Analytic Number Theory
- Chebyshev functions
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abscissa of convergence Blaschke factors Borel-Carathéodory lemma bounds Cauchy's residue theorem Cauchy's theorem characteristic function of divisors Chebyshev function Dirichlet series divisor function entire function Euler's totient function holomorphic function Jensen's inequality Kronecker's lemma Menon's identity Möbius function Picard theorem prime number theorem Ramanujan's sum Riemann zeta function summation by parts units zero-free region
Tag Archives: Cauchy’s residue theorem
Evaluation of integrals using Cauchy’s residue theorem
Let $R$ be a rational function of two variables in $\mathbb{C}$, i.e., $R \in \mathbb{C}(x, y)$. Our goal is to evaluate integrals of the form\[\int_{0}^{2 \pi} R(\cos \theta, \sin \theta) \, d\theta.\] Let $z = e^{i \theta}$. Then\[\cos \theta = … Continue reading