# Contour integral: $\int_{-\infty}^0 \frac{x^{1/3}}{x^5-1}\mathrm dx$

The integral is $$\int_{-\infty}^0 \frac{x^{1/3}}{x^5-1}\mathrm dx$$

I have tried taking the typical half circle contour and finding the enclosed residues: $$\text{Res}(f,1)=\frac{1}{5}\\ \text{Res}(f,e^{\frac{2\pi i}{5}})=\frac{e^{\frac{8\pi i}{15}}}{5}\\ \text{Res}(f,e^{\frac{4\pi i}{5}})=\frac{e^{\frac{-2\pi i}{15}}}{5}$$ By Jordan's lemma, the arc contributes nothing. However, I have a problem with the non integrable singularity at 1 on $\mathbb{R}^+$.

Can I deviate to avoid it while also keeping track of the contribution from the positive real axis? There doesn't seem to be great symmetry at $1$ thanks to the numerator. Is my contour not a good choice?

edit: Also tried changing variables to evaluate on the positive real axis, but this again introduces the bad singularity.

• What's your $x^{1/3}$-definition ?. Without that we don't have any clue about the whole integration meaning. – Felix Marin Sep 27 '17 at 17:50
• @FelixMarin I don't know, it is from a qualification exam I was using to practice. Presumably, if you do valid things given a certain branch cut it is taken to be correct – qbert Sep 27 '17 at 17:51
• I guess the Jack answer assumption $\left(x^{1/3} = \mathrm{sgn}\left(x\right)\left\vert x\right\vert^{1/3}\right)$ is a reasonable def... when we don't have enough information. – Felix Marin Sep 27 '17 at 20:08

Assuming that $x^{1/3}$ is defined as $-(-x)^{1/3}$ on $\mathbb{R}^-$, the change of variable $x=-z^3$ bring the given integral in the form $$\int_{0}^{+\infty}\frac{3z^3}{z^{15}+1}\,dz$$ and by setting $\frac{1}{1+z^{15}}=u$ the previous integral can be computed through Euler's Beta function. By exploiting the reflection formula for the $\Gamma$ function, $\Gamma(z)\,\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$, such integral simplifies to $$\frac{\pi}{5\sin\frac{4\pi}{15}}=\frac{\pi}{5\cos\frac{7\pi}{30}}=\frac{4\pi}{5\sqrt{7-\sqrt{5}+\sqrt{6 \left(5-\sqrt{5}\right)}}}.$$
• @qbert: contour integration works, you just have to "dodge" the singularity at $z=1$ by introducing a suitable deformation of the contour, but the approach above is way more efficient in my opinion. – Jack D'Aurizio Aug 17 '17 at 17:03