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Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$.

The contour is $\gamma=\gamma_1+\gamma_2+\gamma_3$ where $\gamma_1(t)=t$ for $0\leq t \leq R$, $\gamma_2(t)=Re^{i\frac{2\pi}{3}t}$ for $0\leq t \leq 1$, and $\gamma_3(t)=(1-t)Re^{i\frac{2\pi}{3}}$ for $0\leq t \leq 1$.

So, the contour is a wedge, and by letting $R\rightarrow \infty$ we're integrating over one third of the complex plane. I believe this means we are integrating over the entire complex plane under the substitution $u=x^3$. There are poles at $-\zeta$ for each third root of unity $\zeta$, so there's only one pole in this wedge. I'll just refer to that pole as $-\zeta$.

I guess this means that we can use the residue theorem to say $$\int_{\gamma}\frac{1}{1+z^3}dz=2\pi i\eta(\gamma,-\zeta)\operatorname{Res}\left(\frac{1}{1+z^3},-\zeta\right)=2\pi i \lim_{z\rightarrow -\zeta}\left[(z+\zeta)\frac{1}{1+z^3}\right]$$

I can't evaluate this limit. Also I don't see how it involves $R$, which I'm supposed to be taking a limit of. I suspect I've done something wrong.

What's the problem? How do I proceed?

Also, after I do properly evaluate this integral, I am assuming that its value is supposed to be $\displaystyle\int_0^\infty\frac{dx}{1+x^3}$. Why? (I think I know why conceptually but I need to see how one rigorously writes that out.)

(Note: This is exam review, not homework.)

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Samuel Handwich
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3 Answers3

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Hints: Firstly, $1+z^3 = (z+\zeta)(z+\zeta^2)(z+1)$ by factorizing the polynomial.

Secondly, $$\int _0^1\frac{1}{1+z^3} z'(t) \mathrm d t $$ on the contour $z = R(1-t)\exp(2\pi i/3)$ can be related to the real integral you were originally looking for just by substituting this expression for $z$ in.

not all wrong
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The easier way out to compute $\displaystyle \int_0^{\infty} \dfrac{dx}{1+x^3}$ is as follows. We have $$I = \int_0^{\infty} \dfrac{dx}{1+x^3} = \int_{\infty}^0 -\dfrac1{x^2}\dfrac{dx}{1+1/x^3} = \int_0^{\infty} \dfrac{xdx}{1+x^3}$$ We hence have $$2I = \int_0^{\infty} \dfrac{1+x}{1+x^3} dx = \int_0^{\infty}\dfrac{dx}{1-x+x^2} = \int_0^{\infty} \dfrac{dx}{\left(\dfrac{\sqrt3}2 \right)^2+ \left(x - \dfrac12\right)^2}$$ Hence, we get that $$2I = \left. \dfrac2{\sqrt3}\arctan\left(\dfrac{2x-1}{\sqrt3}\right) \right \vert_0^{\infty} = \dfrac2{\sqrt3}\left(\dfrac{\pi}2 + \dfrac{\pi}6\right) \implies I = \dfrac{2 \pi}{3\sqrt3}$$

  • I hate to be picky, but that wasn't the question. I am supposed to be using the contour given. This is for a complex analysis class. Thank you anyway, though, sincerely. (I didn't downvote.) – Samuel Handwich Apr 20 '13 at 23:46
  • @SamuelHandwich No problem. –  Apr 20 '13 at 23:49
  • That is a very nice development. +1 and you're already too veteran for being worried about downvotes from pety people. – DonAntonio Apr 20 '13 at 23:58
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    @DonAntonio :) I am not against down-votes. But down-voting with a reason/ without a comment is what puts me off sometimes. –  Apr 21 '13 at 00:01
  • I know @user17762 :it's annoying sometimes as it can confuse the OP and others about what one did. I almost never downvote (in fact, I don't even remember the last time...), and even if an answer is quite wrong people (1) also learns from mistakes, and (2) we all can write down a comment and tell the answerer what we think before downvoting: what's the rush?! – DonAntonio Apr 21 '13 at 00:05
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    I hate the idea of downvotes without explanation, but if there are two answers, both on 0 votes, say, and one is wrong, I think it's helpful to the OP to distinguish them in this way. I'll remove it if/when it's fixed ofc. – not all wrong Apr 21 '13 at 15:06
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$$\frac{1}{x^3+1}=\frac{1}{3(x+1)}-\frac{x-2}{3(x^2-x+1)}$$

But

$$\frac{x-2}{x^2-x+1}=\frac{1}{2}\frac{2x-1}{x^2-x+1}-\frac{\frac{3}{2}}{\frac{3}{4}+\left(x-\frac{1}{2}\right)^2}=\frac{(x^2-x+1)'}{x^2-x+1}-\frac{4}{3}\frac{\frac{3}{2}}{1+\left(\frac{2}{\sqrt3}\left(x-\frac{1}{2}\right)\right)^2}=$$

$$=\frac{(x^2-x+1)'}{x^2-x+1}-\sqrt3\,\frac{\frac{2}{\sqrt3}dx}{1+\left(\frac{2}{\sqrt3}\left(x-\frac{1}{2}\right)\right)^2}$$

Finally:

$$\int\limits_0^\infty\frac{dx}{x^3+1}=\left.\left[\frac{1}{3}\log\frac{\sqrt{x^2-x+1}}{x+1}+\sqrt 3\arctan\frac{2}{\sqrt 3}\left(x-\frac{1}{2}\right)\right]\right|_0^\infty=$$

$$0+\sqrt3\,\left(\frac{\pi}{2}-\arctan\left(-\frac{1}{\sqrt3}\right)\right)=\sqrt3\left(\frac{\pi}{2}+\frac{\pi}{6}\right)=\frac{2\pi}{\sqrt3}$$

DonAntonio
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