Let the incomplete $\Gamma$-function be defined as: \begin{equation} \Gamma(c+1,x)=\int^\infty_x t^ce^{-t}dt \end{equation} Is there a well studied generalisation of the incomplete gamma function that would work when the R.H.S is \begin{equation} \int^\infty_x f(t)\ e^{-t}dt \end{equation}
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2Consider the Laplace transform of $f$ with a time delay. – Aug 10 '18 at 11:46
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1Is your function has the Laplace transform ? – Nosrati Aug 10 '18 at 11:48
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3If you have no constraints on $f$ at all, then a general theory would not be able to say anything very interesting, because it would need to handle every $\int_x^\infty g(t),dt$ by setting $f(t)=g(t)e^t$. – hmakholm left over Monica Aug 10 '18 at 12:16