All Questions

Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that $$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$ for all $\lambda > 0$. Does it follow that $...

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\...

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...

Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...

I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral: $$\frac{1}{\pi}...

Is there a formula of this integral $$\int_{S_{2n-1}} \frac{e^{i a \langle z, \zeta \rangle}}{|z - \zeta|^{2\lambda}} \, d\sigma(\zeta)$$ and how to calculate it. Thank you in advance