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$\displaystyle \int_{0}^{1} \frac{1}{(x^2-x^3)^{1/3}} \ dx $

I'm having a difficult time understanding how we construct $\displaystyle f(z) = \frac{1}{(z^2-z^3)^{1/3}}$ so that it has a branch cut on $[0,1]$. And then once constructed, how we determine the value of $f(z)$ just above and just below the cut.

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2 Answers 2

up vote 3 down vote accepted

Let us analyze the function for each of the branch points $z=0$ and $z=1$ separately:

  • Around $z=0$, the function $$f_0(z) = z^{-2/3} + O(z^{1/3}).$$ We choose the branch cut such that it extends to $z>0$. Thus, we have $$f_0(z) = |z|^{-2/3} e^{-i (2/3)\arg z} + O(z^{1/3})$$ with $0 <\arg z < 2\pi$. Just above the the branch cut (at $z=0^+ + i 0^+$), we have $\mathop{\rm arg} z = 2\pi$ and thus $f_0(z) = |z|^{-2/3} e^{-i 4\pi/3} = |z|^{-2/3} e^{i 2\pi/3}$. Just below the branch cut (at $z=0^+ - i 0^+$), we have $\arg z =0$ and thus $f_0(z) = |z|^{-2/3} .$

  • At $z=1$, we have a somewhat simpler structure. We can write the function $$f_1(z) = |z^2- z^3|^{-1/3} e^{-i(1/3) [\arg (z^2 -z^3) + 2\pi n]}$$ which has a branch cut along the real line for $z<1$. Here, $n\in\mathbb{Z}$ will be chosen below such that the two expressions $f_0$ and $f_1$ coincide for $z$ close to 0.

So, let us see what happens at $z=0$:

  • Just above the branch cut, the function $f_1(z)$ assumes the form $$f_1(x+i 0^+) = |x^2 - x^3|^{-1/3} e^{-i 2\pi n/3} $$

  • Just below the branch cut, the function $f_1(z)$ assumes the form $$f_1(x-i 0^+) = |x^2 - x^3|^{-1/3} e^{-i 2\pi/3 - i 2\pi n/3} $$

We observe that for $n=-1$ we have found a consistent branch cut structure, where the branch cut only extends between $z=0$ and $z=1$.

The function just above the branch cut reads $$f(x + i 0^+) = | x^2 -x^3|^{-1/3} e^{i 2\pi /3} .$$ Just below the branch cut, we have $$f(x - i 0^+) = | x^2 -x^3|^{-1/3} .$$ For $x>1$, we have $$f(x)= f_1(x) = |x^2 -x^3|^{-1/3} e^{-i(1/3)(\pi -2\pi)} = |x^2 -x^3|^{-1/3} e^{i \pi/3}.$$ For $x<1$, we have $$f(x)= |x^2 - x^3|^{-1/3} e^{-i 2\pi/3}.$$

In general for $|z| \to \infty$ the function assumes the form $$f(z) = |z|^{-1} e^{-i\arg (z)+i \pi/3} = \frac{e^{i \pi/3}}{z}$$ which gives you the residue $e^{i\pi/3}$ at infinity.

Thus, we have $$\int_{0}^{1} [f(x - i 0^+) - f(x + i 0^+) ] \ dx = -2 \pi i e^{i \pi/3}$$ or in other words $$\int_{0}^{1} \frac{1}{(x^2-x^3)^{1/3}} (1- e^{i 2\pi/3}) \ dx = -2 \pi i e^{i \pi/3}$$ from which you obtain (e.g., taking the imaginary part) the result $$\int_{0}^{1} \frac{1}{(x^2-x^3)^{1/3}} = \frac{2\pi}{\sqrt{3}}.$$

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Would you be so kind as to add a simple picture or two? It would make your exelent answer even better =) –  N3buchadnezzar Mar 25 '12 at 21:13
    
@N3buchadnezzar: I am very bad at drawing by hand and even worse drawing with a computer. Thus, I do not think a drawing of mine would help the answer. –  Fabian Mar 26 '12 at 7:23

We can write $f(z) = \alpha z^{-1} (1 - 1/z)^{-1/3}$ where $\alpha$ is one of the cube roots of $-1$, and use the principal branch of $w \to w^{-1/3}$: this has a branch cut for $w$ on the negative real axis, which corresponds to $z \in (0,1)$.

Now for $z = s + i \epsilon$ with $0 < s < 1$ and $\epsilon \to 0+$, $1-1/z \sim 1-1/s + i \epsilon/s^2$ approaches $1-1/s$ from the upper half plane, and $(1-1/z)^{-1/3} \to (1/s-1)^{-1/3} e^{-\pi i/3}$. If you want real values for $f(z)$ as $z$ approaches the branch cut from the upper half plane, you therefore want to take $\alpha = e^{\pi i/3}$. On the other side, for $z = s - i \epsilon$, $1 - 1/z \sim 1-1/s - i \epsilon/s^2$, $(1-1/z)^{-1/3} \to (1/s-1)^{-1/3} e^{+\pi i/3}$, and $f(z) \to (1/s-1)^{-1/3} e^{2\pi i/3}$.

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I don't quite understand why $(1-1/z)^{-1/3} \to (1/s-1)^{-1/3} e^{-\pi i/3}$ from the upper half plane, and why $(1-1/z)^{-1/3} \to (1/s-1)^{-1/3} e^{+\pi i/3}$ from the lower half plane. –  Random Variable Mar 26 '12 at 2:53
    
I would seem to me that $(1-1/z)^{-1/3} \to (1-1/s)^{-1/3} e^{+\pi i/3}$ from the upper half plane. –  Random Variable Mar 26 '12 at 3:24
    
For example if $s = 1/2$, $1/z = 1/(1/2 + i\epsilon) = 2/(1 + 2 i \epsilon) \approx 2 (1 - 2 i \epsilon) = 2 - 4 i \epsilon$ so $1 - 1/z \approx -1 + 4 i \epsilon$. This is in the upper half plane, so for principal-branch powers the $-1$ is treated as $e^{i\pi-}$, and $(1-1/z)^{-1/3} \approx (e^{i\pi-})^{-1/3} \approx e^{-i\pi/3}$. –  Robert Israel Mar 26 '12 at 18:06

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