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Consider the following two polynomials:

f1[t_] := t^3 - 1
f2[t_] := t^3 + 3t - 1

Both of these polynomials have a single real root: $f_1(t)$ at $t = 1$, and $f_2(t)$ at $t \approx 0.322$. Both also satisfy $f'(t) > 0$ at all points in the interval. Let's now try to do the integrals $$ I_1 = \int_0^x \frac{f_1'(t)}{f_1(t)^2} dt \qquad \text{and} \qquad I_2 = \int_0^x \frac{f_2'(t)}{f_2(t)^2} dt $$ for each of these polynomials. (The closed-form indefinite integrals exist in both cases and are $-1/f_1$ and $-1/f_2$ respectively.) We obtain:

integral1 = Integrate[D[f1[t],t] / f1[t]^2, {t, 0, x}]

(* ConditionalExpression[3 (-(1/3) + 1/(3 - 3 x^3)), Re[x] <= 1 || x ∉ Reals] *)

integral2 = Integrate[D[f2[t], t] / f2[t]^2, {t, 0, x}]

(* -1 + 1/(1 - 3 x - x^3) *)

For the first integral, Mathematica correctly recognizes that the integral only converges if $x < 1$. However, it does not seem to be able to recognize that the second integral only converges if $x \lesssim 0.322$.

Is this a bug? Is there a way to tell Integrate to apply additional transformations that might lead it to recognize the conditional convergence of $I_2$ in the same way that it does for $I_1$?

To show that this could lead to unexpected behavior, note that if we set x -> 2 in the above integrals we get

integral1 /. x -> 2

(* Undefined *)  

integral2 /. x -> 2

(* -(14/13) *)

But the integrand of $I_2$ is manifestly positive, and it shouldn't be possible to integrate a positive function to get a negative number. Obviously this is due to the extension of this integral past its domain of applicability; it would be nice if we could somehow modify Mathematica's behavior to tell us that we've done so in both of these cases, not just the case of $I_1$.

I am running Mathematica 10.4.1 on Mac OS X.

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  • $\begingroup$ This question was inspired by this question over on Math StackExchange. $\endgroup$ May 15, 2017 at 18:30
  • $\begingroup$ If you take a complex path around the singularity, it seems to work: Block[{x = 2.}, {NIntegrate[D[f2[t], t]/f2[t]^2, {t, 0, I, x + I, x}], -1 + 1/(1 - x (3 + x^2))}] -- Oddly, adding assumptions that x is real leads to the condition x > 0. $\endgroup$
    – Michael E2
    May 15, 2017 at 19:07
  • $\begingroup$ Presumably any polynomial which does not obviously factor will have that problem... $\endgroup$
    – Igor Rivin
    May 16, 2017 at 3:37

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