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I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula:

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

where, a, b, c presents constant respectively, and $\operatorname{erf}$ presents the error function integral. The exact expression of $\operatorname{erf}$ is $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-\eta^{2}} d \eta$

Thanks a lot.

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    $\begingroup$ If you know the result contains a Bessel function, that seems to imply you've already done some computations that you're not telling us about - how about elaborating? And why are you saying $f$ depends on $\theta $? $\endgroup$
    – Michael Engelhardt
    Commented Aug 3, 2019 at 4:23
  • $\begingroup$ I’m really sorry that I’ve made a mistake that is very obvious. Its calculation doesn’t involve Bessel function directly instead of error function. Only on the condition that b equals 0 and c equals 0, its calculation would involve Bessel function. What’s more, the correct formula will be: $f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) e r f(b \cos \theta+c)$ I’ve modified the question and thanks to your suggestion. $\endgroup$
    – Knight Wang
    Commented Aug 4, 2019 at 3:20
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    $\begingroup$ Also may be written as $$f(a,b,c) = \int_{-1}^1\frac{e^{ax}\mathrm{erf}(bx+c)}{\sqrt{1-x^2}}\;dx$$ $\endgroup$
    – Gerald Edgar
    Commented Aug 4, 2019 at 8:15

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