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Find the integral of $$\int_0^1 \frac{dx}{\sqrt{x^2 - 1}}$$ by considering a dumbbell contour and finding the residue of the branch of $$\frac{1}{\sqrt{z^2 - 1}}$$ at $\infty$.

Now the dumbbell contour consists of a small circle about $z=-1$ (oriented anticlockwise), a small circle about $z=1$ (oriented anticlockwise) and two line segments joining the circles: one above the real axis (directed from $1$ to $-1$) and one below (directed from $-1$ to $1$).

Apparently, the answer is $\dfrac{-\pi \cdot i}{2}$ or $\dfrac{\pi \cdot i}{2}$ depending on what branch is used.

I have been struggling with this integral for some time so if someone could provide a fairly detailed answer, I would be very grateful.

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Let's pick a branch cut on $\Re(z)\in [-1,1]$ such that $\arg(z-1) \in (-\pi,\pi]$ and $\arg(z+1)\in (-\pi,\pi]$. Under this branch cut, we have a continuous function for $x <-1$ $$f(z) = (|z+1|e^{\pm i\pi}|z-1|e^{\pm i\pi})^{-1/2} = \frac{-1}{\sqrt{|z+1||z-1|}} $$

With that in mind, the contour consists of two half-circles and two line segments $$ \begin{align} &C_1: z = -1 + re^{i\theta_1}, \ \theta_1 \in (\pi/2,\pi) \cup (-\pi, -\pi/2] \\ &C_2: z = x + ri, \ x \in (-1,1) \\ &C_3: z = 1 + re^{i\theta_2}, \ \theta_2 \in (-\pi/2,\pi/2) \\ &C_4: z = x - ri, \ x \in (-1,1) \end{align} $$

in the limit $r\to 0$. The integral looks like $$ \int_C = \int_{C_1} + \int_{C_2} + \int_{C_3} - \int_{C_4} $$

Using the estimation lemma, you can prove that both half-circles go to $0$ $$ \left|\int_{C_1,\ C_3} \frac{1}{\sqrt{z^2-1}} dz\right| \le \frac{\pi r}{\sqrt{r^2+1}} \to 0 $$

For the two line segments, observe that $$ f(C_2) \to (|z+1|e^{i0}|z-1|e^{i\pi})^{-1/2} = \frac{-i}{\sqrt{|z+1||z-1|}} $$ $$ f(C_4) \to (|z+1|e^{i0}|z-1|e^{-i\pi})^{-1/2} = \frac{i}{\sqrt{|z+1||z-1|}} $$

Thus $$ \int_C f(z)\ dz \to -2i\int_{-1}^1 \frac{1}{\sqrt{|x^2-1|}}\ dx = -4 \int_0^1 \frac{1}{\sqrt{x^2-1}}\ dx $$

To finish off the integral, we find the residue of $f$ at infinity, which is the same as finding the residue at $0$ of $$ g(w) = -\frac{1}{w^2}f\left(\frac{1}{w}\right) = -\frac{1}{w\sqrt{1-w^2}} $$

You can tell this is a single pole, so $$ \operatorname*{Res}_{w=0} g(w) = \lim_{w\to 0} \big(w g(w)\big) = -1 $$

Altogether this gives a result of $$ \int_0^1 \frac{1}{x^2-1}\ dx = -\frac{1}{4}(-2\pi i) = \frac{\pi i}{2} $$

for this branch


For the other branch, the branch cut is on $\Re(z) \in [1,\infty) \cup (-\infty,-1]$, such that $\arg(z+1)\in (-\pi,\pi]$ and $\arg(z-1) \in [0,2\pi)$. Of course this forces you to invert the contour entirely so the two line segments are parallel to the branch cut, so it may not be as straightforward

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  • $\begingroup$ Hi Dylan, thank you so much for your response, but could you please explain why the argument for $|z+1|$ is 0 for both line segments yet the argument for $|z-1|$ changes from $-\pi$ to $\pi$ $\endgroup$ – Ditherer Dec 14 '17 at 9:28
  • $\begingroup$ I think I'm getting it now, you could also have the argument for $|z-1|$ is 0 for both line segments yet the argument for $|z+1|$ changes from $\pi$ to $-\pi$ $\endgroup$ – Ditherer Dec 14 '17 at 9:44
  • $\begingroup$ Okay, I'm nearly there now... just explain to me a couple of things. The first is the numerator of the bound, $\sqrt{r^2 - 1}$. The second thing is how $ -2i\int_{-1}^1 \frac{dx}{\sqrt{|x^2 - 1|}}$ becomes $-4\int_0^1 \frac{dx}{\sqrt{x^2 - 1}}$. I think the first integral is due to it being the sum of $C_1$, $C_2$, $C_3$, $C_4$. As for the second, the integral from -1 to 1 is twice that from 0 to 1, but how does $i$ cancel? $\endgroup$ – Ditherer Dec 14 '17 at 9:59
  • $\begingroup$ Because $x^2 - 1 < 0$ in the interval so $|x^2-1| = -(x^2-1)$. Also the function is even $\endgroup$ – Dylan Dec 15 '17 at 12:27
  • $\begingroup$ For the other question, imagine that $\arg(z-1)$ and $\arg(z+1)$ each have individual branch cuts in their negative real parts. However, the overlap cuts "cancel" each other out to make their product continuous on $x < -1$. As a consequence of this $\arg(z+1)$ is continuous on $x > -1$ but has a jump on $x < -1$ $\endgroup$ – Dylan Dec 15 '17 at 12:32
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Let $$ f(z)=\frac{1}{\sqrt{z^2-1}}, \quad z\in\mathbb C\setminus[-1,1]. $$ First choose the branch with $f(2)=\frac{1}{\sqrt{3}}$. (You can choose the opposite one too.)

Next observe (not trivial) that $f$ is odd and we have the Laurent series $$ f(z)=\frac{1}{z}+{\mathcal O}(z^{-3}), $$ and hence $$ \int_C f(z)\,dz=2\pi i, $$ where $C$ is the dumbell contour (or any closed simple curve containing $[-1,1]$ in its interior).

In particular, $$ \int_C\frac{dz}{\sqrt{z^2-1}}=\lim_{t\searrow0}\left(\int_{-1}^1\frac{dx}{\sqrt{(x-it)^2-1}}-\int_{-1}^1\frac{dx}{\sqrt{(x+it)^2-1}}\right) \\ \overset{\text{$f$ odd}}{=}2\lim_{t\searrow0}\int_{-1}^1\frac{dx}{\sqrt{(x-it)^2-1}} =4\lim_{t\searrow0}\int_{0}^1\frac{dx}{\sqrt{(x-it)^2-1}}. $$

Finally, $$ \int_0^1\frac{dx}{\sqrt{x^2-1}}=\frac{1}{4}\int_C f(z)\,dz=\frac{\pi i}{2}. $$

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  • $\begingroup$ Sorry, could you provide more details about the Laurent series as well as the very last step? $\endgroup$ – Ditherer Dec 13 '17 at 11:41

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