Is there a solution for the following integral:
$$ \int\frac{\exp(-b(a+x)^{3/2})}{\sqrt{x}} dx $$
where $a$ and $b$ are constants. If it is not, what is the best approximation? Especially in the limit as $b\to\infty$.
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Shouldn't the $dx$ be written at the end? – R_D Nov 05 '15 at 04:53
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Approximation in what limit? As $x \to 0$ or $ \infty$, , as $a \to 0$ or $\infty$, as $b \to 0$ or $\infty$, ...? – Robert Israel Nov 05 '15 at 06:08
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In the limit of $b \to \infty$. – user287139 Nov 06 '15 at 09:31
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If $a=0$, there is an analytical solution for the general integral $$\int \frac{\exp(-p\, x^q)}{x^r}dx=-\frac{x^{1-r} \left(p x^q\right)^{\frac{r-1}{q}} }{q}\,\Gamma \left(\frac{1-r}{q},p x^q\right)$$ For $a\neq0$, I am quite skeptical that we could find any.
Claude Leibovici
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Hint:
$\int\dfrac{e^{-b(a+x)^\frac{3}{2}}}{\sqrt{x}}dx$
$=\int2e^{-b(x+a)^\frac{3}{2}}d(\sqrt{x})$
$=\int2e^{-b(u^2+a)^\frac{3}{2}}du$ $(\text{Let}~u=x^2)$
$=\int\sum\limits_{n=0}^\infty\dfrac{2b^{2n}(u^2+a)^{3n}}{(2n)!}du-\int\sum\limits_{n=0}^\infty\dfrac{2b^{2n+1}(u^2+a)^{3n+\frac{1}{2}}}{(2n+1)!}du$
Harry Peter
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