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I am trying to solve this integral

$$ \int_{20}^{21}\int_{20}^{25}\frac{1}{\sqrt{2\pi}ga_{m}}\exp\Big{(}-\frac{1}{2} \frac{(a_{m}-(ba_{f}+c))^{2}}{g^{2}a_m^{2}}\Big{)}da_{m}da_{f} $$ with b,c,g costants. Do you think is possible to find a closed solution? I have also tried with a symbolic computation with matlab and maxima but it didn't give me any results.

Any clue?

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  • $\begingroup$ Are you using $d$ as a constant here? The presence of $da_m$ in the integrand is potentially confusing. $\endgroup$ – Paul Castle Jun 24 '16 at 14:24
  • $\begingroup$ Yes, yes d, b and c are constants! basically, the integrand is a normal density with mean=$ba_{f}+c$ and standard deviation $da_{m}$ $\endgroup$ – CrishaD Jun 24 '16 at 14:29
  • $\begingroup$ But you're integrating with respect to $a_m$ and $a_f$, is that right? So you're using $d$ to mean two different things. $\endgroup$ – Paul Castle Jun 24 '16 at 16:13
  • $\begingroup$ Ah yes, sorry! Now I got what you meant.. sorry, I will edit the question! $\endgroup$ – CrishaD Jun 24 '16 at 16:15
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With some simple substitutions you can reduce it to

$$\int \int \frac{1}{x} \exp \left( -\frac{(x-y)^2}{x^2} \right) dx dy$$

If you switch the order and integrate with respect to y first, you get something similar to (up to some constants) $$\int \text{erf} \left(\frac{x-a}{x} \right) dx = \int \text{erf} \left(1 - \frac{a}{x} \right) dx$$

This doesn't seem to have a closed form solution in terms of special functions, or at least Mathematica can't find one.

So I'm fairly sure the answer is no. You can still get a numerical answer of course.

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