I am trying to integrate a formula, but I can’t figure out how to do it. Its integrand involves a special function — erf, the error function.

Here's the formula:

$$ f(a, b, c) = \int_{0}^{+\pi} \mathrm d \theta \exp (a \cos \theta) \operatorname{erf}(b \cos \theta+c) $$

where, $ a, b, c $ represent constants and $ \mathrm{erf} $ represents the error function which is expressed by

$$ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} \mathrm e^{-\eta^{2}} \mathrm d \eta $$


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  • 8
    $\begingroup$ Welcome to MMA.SE! It is not possible to have this integral in closed symbolic form, but you can use numeric integration like this: f[a_?NumericQ,b_?NumericQ,c_?NumericQ]:=NIntegrate[Exp[a Cos[\[Theta]]] Erf[b Cos[\[Theta]] + c], {\[Theta], 0, \[Pi]}], then f[1,2,3] gives 3.94476. $\endgroup$ – Alx Aug 5 at 3:13

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