Can the integral $$\int\limits_0^\infty\frac{e^{-t}}{e^{c\sqrt t}-1}dt$$ be evaluated in closed form using some known special functions? ($c \in \Bbb C$)
This is taken from a question on MathOverflow, which aims a finding a function based on its asymptotic expansion. It appears to boil down to solving the above integral.