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I am trying to calculate the integral for $$\int_{-\infty}^{\infty} \frac { e^{-ax^2}}{x^2+\frac{1}{(2a)}}dx$$ I have no idea how to start on this and I have tried Wolfram alpha and similar sites ..the calculation time exceeds the usual time.

I am thinking of taking the derivation of exponential and integrate 1/() term but that would include logarithms and would make things more difficult and if I try to integrate exponential then I would have 1/()^2 term in next step which would again make it difficult ..I really don't know how to go on with this.

I think even a starting step on how to go about this would be a great help.

Gary
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Curiously Curious
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  • This question may be a duplicate of https://math.stackexchange.com/questions/1973280/show-frac2-pi-mathrmexp-z2-int-0-infty-mathrmexp-z2x?noredirect=1 – Maximilian Janisch Feb 14 '22 at 09:30

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We have $$ \int_{ - \infty }^{ + \infty } {\frac{{e^{ - ax^2 } }}{{x^2 + \frac{1}{{2a}}}}dx} = \sqrt {2a} \int_{ - \infty }^{ + \infty } {\frac{{e^{ - \frac{1}{2}t^2 } }}{{t^2 + 1}}dt} = 2\sqrt {2a} \int_0^{ + \infty } {\frac{{e^{ - \frac{1}{2}t^2 } }}{{t^2 + 1}}dt} = \pi \sqrt {2a} e^{\frac{1}{2}} \operatorname{erfc}\left( {\frac{1}{{\sqrt 2 }}} \right) $$ where $\operatorname{erfc}$ is the complementary error function. Note that $$ \pi \sqrt {2a} e^{\frac{1}{2}} \operatorname{erfc}\left( {\frac{1}{{\sqrt 2 }}} \right) = (2.324323459\ldots)\times \sqrt{a}. $$

Gary
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