Let $f(x) = \int_0^\infty g(x,t)dt$, then how can we find the number of zeros (or an upper bound for the number of zeros) of f(x)?
For example,
What about $f(x)=\int_0^\infty e^{-t}t^{x}\sum_1^n c_{i}t^{\mu_{i}} dt$?,
where $c_{i}$ are non zero real numbers and $0< \mu_{1}<\ldots<\mu_{n}$ are positive real numbers.