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I am working on a project on accurate numerical quadrature where I need to compute the following integral in order to find my quadrature weights, $$ \int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\...

Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$ Can we demonstrate that following integral is ...

I am looking for help with doing the following integral : $$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in \...