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I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to reduce such integrals to a finite set of ``master'' integrals which can then be handled case-by-case.

More precisely, I'm interested in integrals of the type

$$\int_0^\infty \frac{P(x,\exp(-x))}{Q(x,\exp(-x))} dx$$

where $P$ and $Q$ are polynomials in two variables (such that the integral exists). The above definite integral will be a function of the parameters of the two polynomials. More precisely, if

$$P(x,y) = \sum_{ij} P_{ij} x^i y^j$$ $$Q(x,y) = \sum_{ij} Q_{ij} x^i y^j$$

then the above integral will be a function $f(P_{ij}, Q_{ij})$. What I'm interested in is the class of functions $f$.

Are there any general statements on integrals of the above type?

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    $\begingroup$ The integrals you wrote are numbers, not functions. What "class of functions" are you talking about in the first line? $\endgroup$
    – Alexandre Eremenko
    May 11, 2016 at 19:52
  • $\begingroup$ @AlexandreEremenko but could that number be calculated (at least in principle) without knowing the closed form of the indefinite integral, maybe utilizing function theory? $\endgroup$
    – Manfred Weis
    May 12, 2016 at 6:07
  • $\begingroup$ I guess you mean "functions that may appear IN definite integrals" instead of "as definite integrals". Please make the wording clearer. Is your idea to find some sort of recursion formula which allows to reduce such an integral to a similar one with e.g. $P$ and $Q$ of smaller total degrees? And before that: have you found any necessary or sufficient conditions on $P$ and $Q$ for the integral to exist? $\endgroup$
    – Wolfgang
    May 12, 2016 at 8:28
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    $\begingroup$ Made the wording more precise. $\endgroup$
    – Fetchinson0234
    May 12, 2016 at 9:39
  • $\begingroup$ Here is a related question: mathoverflow.net/questions/226802/… $\endgroup$
    – Alexandre Eremenko
    May 12, 2016 at 13:16

1 Answer 1

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In the case when there is no exponential in the denominator, the integrand is $$f(x)=\sum_{j=1}^n R_j(x)e^{-jx},$$ where $R_j$ are rational. Then you can decompose the $R_j$ into partial fractions. Integrals $$\int x^ne^{-kx}dx$$ are elementary when $n\geq 0$, while when $n<0$ they are expressed in terms of the logarithmic integral $$\mathrm{Li}(z)=\int_1^ze^x\frac{dx}{x}.$$ So only one function, the logarithmic integral is necessary to evaluate all your integrals in this special case.

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