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

Let $C$ be the ellipse $9x^2+4y^2=36$ traversed once in the counterclockwise direction. Define the function $g$ by $$g(z)=\int_{C}\frac{s^2+s+1}{s-z}ds.$$

Find $g(4i)$.

Well I know I must find $g(z)$ (that is the integral) before computing $g(4i)$, so I decided to use Cauchy's integral formula $f(z_{0})=\frac{1}{2\pi i}\int_C\frac{f(z)}{z-z_{0}}dz$. This put me into trouble, because I do not how to start. Please i need a hint.


share|cite|improve this question
Try putting in the $4i$ first, then evaluating the integral. It's a lot easier this way because then you're working with a specific integral. – Ted Mar 24 '12 at 5:40
@Hassan To make it more familiar, start with changing $z$ with $z_0$ and $s$ with $z$. – Pedro Tamaroff Mar 24 '12 at 5:41 I will try it. – Hassan Muhammad Mar 24 '12 at 5:58
Also you should remember if f(z)/(z-z_0) is analytic within and on C, then the integral is 0. – Apprentice Queue Mar 24 '12 at 6:09


  1. Carefully sketch the curve $C$.
  2. Use some theorem of Cauchy.

It might also be interesting to look at other points than $z=4i$, e.g. $z=0$...

share|cite|improve this answer
What do you mean by saying 'use some theorem of cauchy?' More clarifications please. – Hassan Muhammad Mar 24 '12 at 6:15
Use the one used in the proof of the Cauchy integral formula... – AD. Mar 24 '12 at 7:15
Ok, $g(0)=1$, $g(i)=i$ Correct? But I think $4i$ is outside the ellipse. – Hassan Muhammad Mar 24 '12 at 7:47
Yes, $4i$ is outside, hence the curve is contractable to a point.. – AD. Mar 24 '12 at 10:31
Hassan, you should look closer to Cauchy's integral theorem... – AD. Mar 24 '12 at 11:58

$$g(4i)=\int_C \frac{s^2+s+1}{s-4i}ds$$

$\frac{x^2}{4}+\frac{y^2}{9}=1$ is the equation of the vertical ellipse with vertices at $-2, 2, -3i, 3i$. Therefore $4i$ is outside the ellipse and the function $f(s)=\frac{s^2+s+1}{s-4i}$ is analytic within $C$. Then the integral is $0$.

share|cite|improve this answer
What is $g(i)$? – Hassan Muhammad Mar 27 '12 at 6:06
$g(i)=\int_C \frac{s^2+s+1}{s-i}ds$. In this case, $i$ is within the ellipse and we assume $f(s)=s^2+s+1$. using Cauchy Integral Formula, we get $g(i) = 2\pi i f(i) = -2\pi$. – MNos Apr 1 '12 at 14:02

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.