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I am calculating the following integral $$I=\int_0^1 e^{-c_1x^2+c_2x}\mathrm{erf}(c_3x+c_4)dx,$$ with Integrate and NIntegrate respectively.

There is no analytical solution for this integral and I got two different numerical results, with quite large difference: $-0.117035 + 0.5i$.

My question is, which one is more accurate in this case.

Here is my code:

c1 = Pi;
c2 = SetPrecision[10.1 + 10.1 I, 30];
c3 = Sqrt[Pi];
c4 = SetPrecision[1.1 + 6.1 I, 30];
I1 = NIntegrate[Exp[-c1 x^2 + c2 x] Erf[c3 x + c4], {x, 0, 1}, 
   PrecisionGoal -> 20, WorkingPrecision -> 30];
I2 = Integrate[Exp[-c1 x^2 + c2 x] Erf[c3 x + c4], {x, 0, 1}];
error = N[I1 - I2]
(*error=-0.117035 + 0.5i*)

Thanks in advance!!

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1 Answer 1

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Better define the parameters with infinite precision. Then you can use ultrahigh WorkingPrecision.

c1 = Pi;
c2 = (101 + 101 I)/10;
c3 = Sqrt[Pi];
c4 = (11 + 61 I)/10;

I3 = NIntegrate[Exp[-c1 x^2 + c2 x] Erf[c3 x + c4], {x, 0, 1}, 
         WorkingPrecision -> 100]

(*   5.98422153733426875937314115532253471792710605523929063484400170963328\
        0746439954199937370126209185221*10^10 - 
     4.7006274392441169485465943452912663047232799365447530746425809872967\
        02374014674651258602665889195814*10^13 I   *)
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  • $\begingroup$ @Akkr14: thanks for the answer^_^! After Rationalize the parameters, I recalculated $I_1$, $I_2$ and $I_3$. It turned out that $I_1$ and $I_3$ via NIntegrate are almost the same, but $I_2$ is still much different than them. Does that mean the result obtained by Integrate is not trustable in this case? $\endgroup$
    – Ellery
    Feb 28, 2020 at 9:52
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    $\begingroup$ Remove the semicolon after Integrate... to see, that it gives back the input. It does not find an analytical solution, most likely the is no one. But when you apply N[I2] the integral is automaticlly calculated numericaly with NIntegrate, but with standard machine precision, which is 16. This is of course less accurate than with WorkingPrecission->30 . $\endgroup$
    – Akku14
    Feb 28, 2020 at 14:33
  • $\begingroup$ @Akkr14: thanks a lot! it is very clear now. $\endgroup$
    – Ellery
    Feb 29, 2020 at 17:11

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