# Integrating an expression containing the error function, perf

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

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

## migrated from mathematica.stackexchange.comAug 8 at 7:54

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• 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. – Alx Aug 5 at 3:13