How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$ Could you explain to me, with details, how to compute this integral, find its principal value?
$$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$
$f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} (z^2-1)}$
It has singularities on the real line, so I need to integrate this function over a curve made of segment $[0, 1- \varepsilon] \cup $ semicircle with endpoints $1- \varepsilon, \ 1+ \varepsilon \cup$ segment $[1+ \varepsilon, R]$ and we link $R$ and $0$ with a big semicircle .
The integral over the big semicircle vanishes.
Integral over the small semicircle, centered in $1$ tends to $i \pi Res_1f$ as its radius tends to $0$.
Could you tell me how to calculate this put this together and find this integral?
Thank you.
 A: The integral as stated does not converge.  On the other hand, its Cauchy principal value exists and may be computed using the residue theorem.
Consider the integral
$$\oint_C dz \frac{\sqrt{z}}{z^2-1} $$
where $C$ is a keyhole contour of outer radius $R$ and inner radius $\epsilon$, with semicircular detours of radius $\epsilon$ into the upper half plane at $z=1$.  The contour integral is then
$$\int_{\epsilon}^{1-\epsilon} dx \frac{\sqrt{x}}{x^2-1}+  i \epsilon \int_{\pi}^0 d\phi \, e^{i \phi} \frac{\sqrt{1+\epsilon e^{i \phi}}}{\left (1+\epsilon e^{i \phi} \right )^2-1} \\  + \int_{1+\epsilon}^R dx \frac{\sqrt{x}}{x^2-1} + i R \int_0^{2 \pi} d\theta \, e^{i \theta} \frac{\sqrt{R} e^{i \theta/2}}{R^2 e^{i 2 \theta}-1} \\ + \int_{R}^{1+\epsilon} dx \frac{e^{i \pi} \sqrt{x}}{x^2-1} + i \epsilon \int_{2\pi}^{\pi} d\phi \, e^{i \phi} \frac{\sqrt{e^{i 2 \pi}+\epsilon e^{i \phi}}}{\left (1+\epsilon e^{i \phi} \right )^2-1} \\ + \int_{1-\epsilon}^{\epsilon} dx \frac{e^{i \pi} \sqrt{x}}{x^2-1} + i \epsilon\int_{2 \pi}^0 d\phi \frac{\sqrt{\epsilon} e^{i \phi/2}}{\epsilon^2 e^{i 2 \phi}-1}$$
As $R \to \infty$, the magnitude of the fourth integral vanishes as $R^{-1/2}$.  As $\epsilon \to 0$, each of the second integral contributes a factor of $-i \pi/2$, the sixth integral contributes a factor of $i \pi/2$ and the eighth integral vanishes as $\epsilon^{3/2}$.  Also, the first, third, fifth, and seventh integrals add to the Cauchy principal value of an integral.  Thus, the contour integral is equal to
$$2 PV \int_{0}^{\infty} dx \frac{\sqrt{x}}{x^2-1} $$
By the residue theorem, the contour integral is equal to $i 2 \pi$ times the residue of the pole $z=e^{i \pi}$ inside $C$.  Thus,
$$PV \int_{0}^{\infty} dx \frac{\sqrt{x}}{x^2-1} = i \pi \frac{e^{i \pi/2}}{2 e^{i \pi}} = \frac{\pi}{2} $$
A: Let $\delta_1>0,\delta_2<1$ real numbers close to zero and one, respectively, and I=$(\delta_1,\delta_2)\cup(\delta_2^{-1},\delta_1^{-1})$.
We have:
$$ \int_{I}\frac{\sqrt{x}}{x^2-1}\,dx = \int_{\delta_1}^{\delta_2}\frac{\sqrt{x}}{x^2-1}\,dx +\int_{\delta_1}^{\delta_2}\frac{\sqrt{1/x}}{1-x^2}\,dx = \int_{\delta_1}^{\delta_2}\frac{1}{x^{1/2}+x^{-1/2}}\frac{dx}{x}$$
and by setting $x=u^2$:
$$ \int_{I}\frac{\sqrt{x}}{x^2-1}\,dx = 2\int_{\sqrt{\delta_1}}^{\sqrt{\delta_2}}\frac{du}{u^2+1}, $$
so, by letting $\delta_1\to 0,\delta_2\to 1$, we get that the principal value of the original integral is given by:
$$ 2\arctan(1) = \color{red}{\frac{\pi}{2}}.$$
A: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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$\ds{\mbox{P.V.}\int_{0}^{\infty}\frac{\root{x}}{x^{2} - 1}\,\dd x:\ {\large ?}}$.

We'll consider the $\ds{\root{z}}$-branch cut:
$$
\root{z}=\verts{z}\exp\pars{\half\,{\rm Arg}\pars{z}\ic}\,,\qquad
0 < \,{\rm Arg}\pars{z} < 2\pi\,,\quad z \not= 0
$$
We'll perform an integration over a key-hole contour $\ds{\gamma}$ which takes care of the above branch-cut:
$$
\oint_{\gamma}\frac{\root{z}}{z^{2} - 1}\,\dd z
=2\pi\ic\bracks{\verts{\expo{\pi\ic}}^{1/2}\exp\pars{\half\,\pi\ic}\,\frac{1}{2\expo{\pi\ic}}}=\pi\tag{1}
$$
The contribution from a 'big arc' of radius $\ds{R}$ goes as $\ds{R^{-1/2}}$
when $\ds{R \to \infty}$ while the contribution of a 'small arc' of radius $\ds{\epsilon}$, around the origin, goes as $\ds{\epsilon^{3/2}}$ when $\ds{\epsilon \to 0^{+}}$.

Then,
\begin{align}
\oint_{\gamma}\frac{\root{z}}{z^{2} - 1}\,\dd z
&=\int_{0}^{\infty}
\frac{\root{x}\expo{0\ic/2}}{\pars{x - 1 + \ic 0^{+}}\pars{x + 1}}\,\dd x
+\int_{\infty}^{0}
\frac{\root{x}\expo{2\pi\ic/2}}{\pars{x - 1 - \ic 0^{+}}\pars{x + 1}}\,\dd x
\\[5mm]&=\int_{0}^{\infty}\root{x}\bracks{%
\frac{1}{\pars{x - 1 + \ic 0^{+}}\pars{x + 1}}
+\frac{1}{\pars{x - 1 - \ic 0^{+}}\pars{x + 1}}}\,\dd x
\\[5mm]&=2\,\mbox{P.V.}\int_{0}^{\infty}\frac{\root{x}}{x^{2} - 1}\,\dd x\tag{2}
\end{align}
With $\pars{1}$ and $\pars{2}$:
$$
\color{#66f}{\large\mbox{P.V.}\int_{0}^{\infty}\frac{\root{x}}{x^{2} - 1}\,\dd x}
=\color{#66f}{\large\frac{\pi}{2}}
$$
A: Here is an approach. Recalling the Mellin transform

$$ F(s) = \int_{0}^{\infty} x^{s-1}f(x)dx.$$

We have the Mellin of our function $f(x)=\frac{1}{x^2-1}$ is given by

$$ F(s) = -\frac{\pi}{2} \cot(\pi s/2). $$

So our integral can be evaluated as

$$ I =\lim_{s\to 3/2}F(s) = \frac{\pi}{2} $$

