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Sir, While, studying the diffraction of the Gaussian beam through apertures, I have faced the following integral (as an expression of diffracted field), $$I= \int_0^R \exp(-\alpha r^2+i\beta r)\bigg[\operatorname{erfi}(A+i B r)-\operatorname{erfi}(A-i B r)\bigg] \,dr$$ where, $\alpha$, $\beta$, $A$ and $B$ are constants.

and I am trying to find out the closed form answer of this integral, from which further analysis can be made by varying/ adjusting the beam parameters following the experimental set up.

Sir, would you kindly suggest me how I can get the closed form answer of this integral or any relevant references from where I can get some help.

Thanking you....

Gary
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R. Bhattacharya
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  • @Gary...Thanks.... – R. Bhattacharya Feb 13 '22 at 06:49
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    Complete the square in the exponent. Split the integral. Use $\operatorname{Erfi}(z)=-i\operatorname{Erf}(iz)$. The resulting integral of error function times Gaussian is evaluated here – Sal Feb 13 '22 at 14:09
  • @Sal... Sir, thanks for your kind response. But I have a doubt that in the link, you have shared, the limit of the integral goes from (- Infinity) to a finite number, where in my case, the limits are finite. Further, I have gone through the solution shown here link but at the final stage, while doing the partial integration I got back the initial expression again. – R. Bhattacharya Feb 13 '22 at 19:02
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    Okay, so $J(\beta)=\int\limits_\beta^\infty \dots$ is the integral in your linked post. The expression for $J$ is in that post too: it's the lengthy one involving $\tan^{-1}$ and $T$, the Owen-T function. Once you have that, you can find the integral with other limits since $\int\limits_a^b=\int\limits_a^\infty-\int\limits_b^\infty$ – Sal Feb 13 '22 at 19:55
  • @ Sal......Sir, Thanks once again...I am trying to solve following your suggestion. If I get stuck with a particular step, I shall disturb you again.... – R. Bhattacharya Feb 14 '22 at 14:39
  • @Sal.......Sir, I am still facing the trouble at the last stage where the constant C [appears in addition to the Owen T function] is to be determined [after equating partial derivative at both sides]. While trying to determine C, I have got back again the integration similar to the initial expression. What can I do now? – R. Bhattacharya Feb 26 '22 at 09:41
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    In the link the constant $C$ is also given in terms of $\alpha$ and $c$ – Sal Feb 26 '22 at 12:37
  • @Sal....Sir, thank you...finally I got it and the Answer is in terms of Generalized Owen T Function...But I think the plot of Generalized Owen T function is not possible in Mathematica , it should be done by integrate numerically..On the other hand, the plot of Owen T function i.e. T(x,a) is available in Mathematica. – R. Bhattacharya Feb 26 '22 at 16:29
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    In the link the result is also given in terms of the regular Owen T function and $\tan^{-1}$ – Sal Feb 27 '22 at 15:07
  • @Sal.....Sir, got it...it is there in the link...Is there any book or any reference where this "Generalized Owen T function" has been dealt with. I have searched through Google, but it only shows the derivation in Mathematics stack exchange. – R. Bhattacharya Feb 28 '22 at 15:09

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