How should I evaluate the numerical integral of a function with divergent part substracted?

Question

I want to evaluate the following integral numerically in Mathematica, $$\int_0^{\infty}(\frac{1}{(1 - e^t)^{10}} - (\frac{1}{t^{10}} - \frac{5}{t^9} + \frac{145}{12 t^8} - \frac{75}{ 4 t^7} + \frac{3013}{144 t^6} - \frac{285}{16 t^5} + \frac{4523}{378 t^4} - \frac{6515}{ 1008 t^3} + \frac{7129}{2520 t^2} - \frac{1}{t}))dt$$ which is convergent since the part that is subtracted from $\frac{1}{(1-e^x)^{10}}$ is essentially the negative powers of the Laurent expansion of $\frac{1}{(1-e^x)^{10}}$ near 0, which is the divergent part.

However, if I directly use NIntegrate in Mathematica, I get a divergent result. This is not too surprising given the unrealistic plot of the integrand near zero:

So is there a good way to actually evaluate this numerical integral?

Answer 1

It seems that your integral effectively diverges. Let's define:

ee[x_] := 1/(1 -E^x)^10
(*div is the divergent part*)
div[x_] := Evaluate[Series[ee[x], {x, 0, -1}] // Normal]
all[x_] := ee[x] - div[x]

You're right that you "killed" the divergence at zero:

Integrate[all[x], {x, 0, 1}] // N
(* 0.253202 *)

But the "original" function's integral was already converging in {1, Infinity}

Integrate[ee[x], {x, 1, Infinity}] // N
(* 0.0002918 *)

While the "new" added part integral diverges in this interval:

Integrate[div[x], {x, 1, Infinity}]

Integrate::idiv: Integral of div[x] does not converge on {1,[Infinity]}. >>

so the whole thing won't converge

As your integral from 1 onwards has a closed (but longish) form, if you want a peep at the divergence form you may evaluate

k = Integrate[all[x], {x, 1, p}, Assumptions -> Element[p, Reals]];
Plot[k, {p, 1, 100}]

and

s = k /. p -> 1/u;
Plot[s, {u, -1, 1}, PlotRange -> {{-1, 1}, {0, 10}}]

Answer 2

I think Taylor expansion is a bad regulator to regulate the divergent part of the integration. The following method is meant to give you a feel of the behavior instead of a concrete solution.

First, rewrite your integration to $$\int_0^\infty dt\, \frac{1}{(1-e^{t+i \epsilon})^{10}}.$$ we will set $\epsilon\to 0$ at last step to see how it behave.

The above integration can be easily done by

Integrate[1/(1 - E^(t + I \[Epsilon]))^10, {t, 0, \[Infinity]}, Assumptions -> \[Epsilon] > 0]

you will get:

$$\frac{num1}{2520 \left(-1+e^{i \epsilon }\right)^9}+\log \left(1-e^{-i \epsilon }\right)$$ $$num1:=-41481 e^{i \epsilon }+120564 e^{2 i \epsilon }-210756 e^{3 i \epsilon }+236754 e^{4 i \epsilon }-173250 e^{5 i \epsilon }+80220 e^{6 i \epsilon }-21420 e^{7 i \epsilon }+2520 e^{8 i \epsilon }+7129$$

the $num1\to 280$ when $\epsilon \to 0$, so Taylor expansion it

Series[280/(2520 (-1 + E^(I \[Epsilon]))^9) + Log[1 - E^(-I \[Epsilon])], {\[Epsilon], 0, 1}]

gives you:

$-\frac{i}{9 \epsilon ^9}-\frac{1}{2 \epsilon ^8}+\frac{13 i}{12 \epsilon ^7}+\frac{3}{2 \epsilon ^6}-\frac{1069 i}{720 \epsilon ^5}-\frac{89}{80 \epsilon ^4}+\frac{29531 i}{45360 \epsilon ^3}+\frac{761}{2520 \epsilon ^2}-\frac{i}{9 \epsilon }-\frac{236250367 i \epsilon }{479001600}+\log (\epsilon )-\frac{2857}{89600}+\frac{i \pi }{2}+O(\epsilon^1)$

Then you can read the singular part.