Can anyone see a solution for varitations of

$$\int \frac{1}{x+b}\exp(-(x\pm b)^2)dx$$

I recognize that the term $1/(x+b)$ could be rewritten for the region $|x|<b$

$$\int \sum_{i=1}^\infty \frac{(-x)^i}{b^i} \exp(-(x \pm b)^2) \textrm{d}x $$

The series can be modified to an arbitrarily large convergence domain by changing the center of the expansion

$$\int \sum_{i=1}^\infty \frac{(x+L)^i}{(L -b)^i} \exp(-(x \pm b)^2) \textrm{d}x $$

But I can't go much further with this approach.

I expect (not required) to have some solution of the form $Q_1(b,x) \textrm{erfc}(x \pm b) + Q_2(b,x) e^{-(x\pm b)^2}$ where $Q_1(x),Q_2(x)$, or a series in terms of $Q_k(x) \gamma_k(x)$ (incomplete gamma function), and the $Q$ functions are rational. Valid for specific intervals that will cover the positive real domain. Or simply an exact series that converges quikly for large $b$.

Thank you

  • 2
    $\begingroup$ For the upper sign, take $\sqrt{t}=x+b$. $\endgroup$
    – Gary
    Apr 4, 2022 at 8:28
  • $\begingroup$ @Gary Thank you, I hope for the lower sign there is anything like that hehe $\endgroup$
    – Bob
    Apr 4, 2022 at 8:33
  • $\begingroup$ I do not want to be pessimistic but, for the lower sign, as Dante Alighieri wrote 700 years ago, for the antiderivative : "O you who enter here give up all hope" $\endgroup$ Apr 4, 2022 at 10:10
  • $\begingroup$ For the definite integral $\int_0^\infty$, there is a simple solution. $\endgroup$ Apr 4, 2022 at 11:54


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