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Doing Integrate[1/(x y),{x,0,1},{y,0,1}] returns zero, although the integral is clearly divergent; NIntegrate attempts to compute it, then gives up at 2.15434*10^9 with some error messages.

Is there a way to get a correct answer?

There are several complaints here about strange evaluations of integrals, but none of them are quite like this. The closest one, Strange result for divergent double integral $\int _0^{\infty }\int _0^{\infty }\frac{1}{x^2 y^2+1}dydx$ does not have any answers, so I decided to post this as another instance of the same problem.

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    $\begingroup$ What MMA version and operating system are you on? I get the same on 10.0.2.0 on macOS 10.12.4 $\endgroup$
    – Marius Ladegård Meyer
    Commented May 6, 2017 at 10:09
  • $\begingroup$ @MariusLadegårdMeyer Mine is 11.0.1.0 on WIndows 10 $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 6, 2017 at 10:48
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    $\begingroup$ Version 10.0 gives the error: "Integral of 1/x does not converge on {0,1}." $\endgroup$
    – mattiav27
    Commented May 6, 2017 at 11:22
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    $\begingroup$ @მამუკაჯიბლაძე I meant that using your function, Mathematica gives correctly an error. I meant it works with my version of Mathematica. $\endgroup$
    – mattiav27
    Commented May 6, 2017 at 12:23
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    $\begingroup$ I will file a bug report but I should mention that any "simple" fix might turn out to break things. $\endgroup$
    – Daniel Lichtblau
    Commented May 8, 2017 at 21:11

2 Answers 2

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This has been fixed as of version 11.3.0:

Integrate[1/(x y), {x, 0, 1}, {y, 0, 1}] // Head

Integrate::idiv: Integral of 1/(x y) does not converge on {0, 1}.

(* Integrate *)
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The limit of (Integrate[1/(x y), {x, a, 1}, {y, a, 1} ], a -> 0) is Infinity.

In:

Clear[n]
f[a_] := Integrate[1/(x y), {x, a, 1}, {y, a, 1} ]
Limit[f[a], a -> 0]
Table[{10^(-n), f[10^(-n)] // N}, {n, 200, 100, -1}] // 
 ListLinePlot[#, PlotRange -> Full] &

Out: enter image description here

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  • $\begingroup$ As a matter of fact, Integrate[1/(x y), {x, a, 1}, {y, a, 1} ] is Log[a]^2 (under some restrictions on a) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 7, 2017 at 5:28

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