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In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2. they note that \begin{align} \frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds = 2^{-\nu/2} \pi^{-...

I brought up a couple of combinatorial and number-theoretic items with this MO question. Now, I shall inquire on growth estimates. Recall $$K_r(n):=\prod_{\ell_1=1}^n\cdots\prod_{\ell_r=1}^n\left( 4\...

Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$: $f(z)$ is bounded when $\Re z>1+\delta$ $f(z)$ is unbounded when $\Re z=1$ $f(z)$ grows polynomially ...

For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k_1 < k_2 <\cdots < k_n < q<2^{2n}$ such that $$\prod_{i=1}^{n} \sin\left(\frac{k_i \pi}{q} \...

I want to know when certain expressions of the form $ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $ are algebraic numbers. These ...