I am trying to find an analytic answer for $\int_b^c\frac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$ but it doesn't seem to be in any of the integral tables that I've looked in. I don't think contour integration will help because there is no way to choose a branch cut that will give me the desired limits. Can someone confirm this?
2026-04-22 02:48:58.1776826138
Is there no analytic form of $\int_b^c\frac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$ ?
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Hint:
$\int_b^c\dfrac{\sqrt{x}e^x\text{erfc}(\sqrt{x})}{\sqrt{a-x}}dx$
$=\int_b^c\dfrac{\sqrt{x}e^x}{\sqrt{a-x}}dx-\int_b^c\dfrac{\sqrt{x}e^x\text{erf}(\sqrt{x})}{\sqrt{a-x}}dx$
$=\int_b^c\sum\limits_{n=0}^\infty\dfrac{x^{n+\frac{1}{2}}}{n!\sqrt{a-x}}dx-\int_b^c\sum\limits_{n=0}^\infty\dfrac{2^{2n+1}n!x^{n+1}}{(2n+1)!\sqrt\pi\sqrt{a-x}}dx$ (according to http://en.wikipedia.org/wiki/Dawson_function)