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A known generalization of Levi-Civita field is a field of Hahn power series of $\varepsilon$ of the form $\mathbb{R}[[\varepsilon^{\mathbb{Q}}]]$. Assuming $\varepsilon=1/\omega$, we can naturally embed a set of divergent integrals in our field as formal power series of $\omega$:

$$F(\omega)=F(a)+\int_a^\infty F'(x)dx$$

This way the ordering of divergent integrals will correspond to the ordering of their growth rates represented as powers series.

Assuming the Levi-Civita type of multiplication operation, we can obtain the multiplication rule for divergent integrals:

$$\int_0^\infty f(x)dx\cdot\int_0^\infty g(x)dx$$ $$=D^{-1}f(x)D^{-1}g(x)|_{x=0}+\int_0^\infty D_x[\int_0^x f(t)dt \int_0^x g(t)dt]dx$$ $$-\operatorname{reg}\int_0^\infty D_x[\int_0^x f(t)dt \int_0^x g(t)dt]dx$$

The $D^{-1}$ is assumed to be natural integral here.

The above formula can be coded in Mathematica system with the following code:

f[x_] := Exp[x]
g[x_] := Exp[x]
prod1[x_] := 
 Evaluate[Refine[Integrate[f[x], x] Integrate[g[x], x], x > 0]]
prod2[x_] := 
 Evaluate[Refine[
   Integrate[f[x], {x, 0, x}] Integrate[g[x], {x, 0, x}], x > 0]]
Inactivate[
    Integrate[f[x], {x, 0, Infinity}]\[CenterDot]Integrate[
      g[x], {x, 0, Infinity}], Integrate] == 
   FullSimplify[
     prod1[0] + 
      Distribute[
       Integrate[
        ExpandAll[FullSimplify[D[prod2[x], x]]], {x, 
         0, \[Infinity]}]]] - 
    Limit[Sum[D[prod2[s x], x], {x, 1, Infinity}, 
       Regularization -> "Dirichlet"] // FullSimplify, s -> 0] // 
  ExpandAll // Quiet
Inactivate[
  Reg[Integrate[f[x], {x, 0, Infinity}]\[CenterDot]Integrate[
     g[x], {x, 0, Infinity}]], Integrate] == FullSimplify[prod1[0]]

For instance, as the program outputs, $$\int _0^{\infty }e^xdx\cdot \int _0^{\infty }e^xdx=\int_0^{\infty } 2 e^{2 x} \, dx-\int_0^{\infty } 2 e^x \, dx.$$

The code has two caveats. First, it assumes the indefinite integrals produced by Mathematica are natural integrals (which is the case for the basic elementary functions). Second, it uses a series regularization, in this case, Dirichlet, but for other functions another method, like Dirichlet may be needed.

As follows from Levi-Civita multiplication, the regularized value of the product of two integrals is the product of the regularized values. Thus, knowing that $\operatorname{reg}\int_0^\infty e^x dx=-1$, we can conclude that this integral squared has the regularized value of $1$.

That said, I have the following questions.

Previously I already tried to define multiplication of divergent integrals in a different way. Now I see that Levi-Civita type of multiplication is the natural way (limit of the product should be equal to the product of limits, etc). For instance, the older approach would give

$$\int_0^\infty (2x^3-3x^2+x-4) \, dx \cdot \int_0^\infty (2x^2-3x+1) \, dx$$ $$= \int_0^\infty \left(\frac{7 x^6}{3}-\frac{17 x^5}{2} + 10 x^4-\frac{41 x^3}{3}+\frac{1007x^2}{60}-\frac{63 x}{10}-\frac{113}{120}\right) \, dx+\frac{127}{420}$$

while Levi-Civita type of multiplication gives for the same integrals

$$\int_0^\infty (2x^3-3x^2+x-4) \, dx \cdot \int_0^\infty (2x^2-3x+1) \, dx$$ $$=\left(\frac{\omega ^4}{2}-\omega ^3+\frac{\omega ^2}{2}-4 \omega \right) \left(\frac{2 \omega ^3}{3}-\frac{3 \omega ^2}{2}+\omega \right)=$$ $$\frac{\omega ^7}{3}-\frac{17 \omega ^6}{12}+\frac{7 \omega ^5}{3}-\frac{53 \omega ^4}{12}+\frac{13 \omega ^3}{2}-4 \omega ^2=\int_0^{\infty } \left(\frac{7 x^6}{3}-\frac{17 x^5}{2}+\frac{35 x^4}{3}-\frac{53 x^3}{3}+\frac{39 x^2}{2}-8 x\right) \, dx$$

But the older approach was while more complicated, at the same time, more interesting because of connection with Bernoulli numbers and Zeta function.

  • As such, I wonder whether the older approach can be somehow embedded in the newer set by choosing a suitable basis of otherwise? Levi-Civita approach reminds me dual numbers while my old approach is similar to split-complex numbers.

  • Can the Levi-Civita multiplication code and the formula be simplified?

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    $\begingroup$ I know that I have made these kinds of comments before, but it really seems like you are trying to work out the details of a research program in real time, and inviting others to join the program with you—which is a great mathematical activity, but seems like it is not the sort of use to which MO is intended to be put. $\endgroup$
    – LSpice
    Jan 15, 2022 at 18:18
  • $\begingroup$ @LSpice is not this site for something like "research questions"? $\endgroup$
    – Anixx
    Jan 15, 2022 at 18:21
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    $\begingroup$ Certainly, this site is for research questions, and any one person's opinion (such as mine) about what is appropriate for it does not mean much. However, my understanding is that "research question" is an upper bound, not a lower bound: only questions about research, broadly understood, are acceptable, but not all such questions are acceptable; the ideal MO question, as I understand it, should have a single, well defined answer. (The existence of big-list and community Wiki problems are a counterexample … I don't propose a rigorous theory here.) $\endgroup$
    – LSpice
    Jan 15, 2022 at 18:24
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    $\begingroup$ @LSpice I will think on how to improve this question so to narrow it down. $\endgroup$
    – Anixx
    Jan 15, 2022 at 18:26

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