Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Could someone help me through this problem?

Let C be an arc of the circle $|z|=R$, with $R>1$ of angle $\frac{\pi}{3}$.

Show that $\left|\displaystyle\int_{C} \frac{1}{z^{3}+1}\, dz\right|\leq \dfrac{\pi}{3}\left(\dfrac{R}{R^{3}-1}\right)$

and deduce $\lim\limits_{R \to{+}\infty}{\displaystyle\int_{C} \frac{1}{z^{3}+1}\, dz}$

share|cite|improve this question
up vote 2 down vote accepted

Hint: For any contour $\Gamma$ one has that

$$\left|\int_\Gamma f(z)\;dz\right|\leqslant \|f\|_\infty|\Gamma|$$

Where $|\Gamma|$ is the arc-length of $\gamma$.

share|cite|improve this answer
As a supplement to this hint, for any complex $a$ and $b$ one has $|a - b| \geq ||a| - |b||$. So in particular, if $|z| = R > 1$, putting $a = z^3$ and $b = -1$ one deduces that $|z^3 + 1| \geq R^3 - 1$. – leslie townes Apr 26 '12 at 3:13

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.