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Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...

Consider the following function: $$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$ Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant. The following three conditions ...

In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function: \begin{equation} \mathrm{D}_{\rm b}(x,n)=\prod_{...

Consider the following function : $$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$ Here, $\omega(z)$ is a weight we are going to consider The following two conditions should meet for $\...

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$. Using maple, it seems that $Re(\dfrac{I'...