Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 1 down vote favorite

Find the saddle points $z_{1},z_{2}$ of $f(z)=\frac{(z - 1)^{2}(z + 1)}{z^{2}}$

Does anyone can help me with this problem?

$z_{0}$ is a saddle point of an analytic function f if and only if $f´(z_{0})=0$ and $f(z_{0})=0$.

share|improve this question
2  
What have you tried? – Antonio Vargas May 4 '12 at 5:36
Maybe you could try looking at the nicely factored formula for $f$ to guess where $f(z)=0$? There are only two such points in $\mathbb{C}$. Then try differentiating the formula using the product and quotient rules. Then check to see if any of the solutions to $f(z)=0$ satisfy $f'(z) = 0$ also. Just as a hint, you can figure all of this out from the formula for $f$ without having to do any work. – copper.hat May 4 '12 at 6:59
@Breton: Saddle point $z_0 $ is where $f'(z_0)=0$ as well as $f''(z_0)=0$. there is problem in your definition – Avatar Jul 12 '12 at 7:49

1 Answer

up vote 1 down vote

I've never seen that definition of a saddle point before. There must be a mistake somewhere, since there's only one such point for that function. A hint for finding it: Before differentiating, multiply through by $z^2$; then the derivative of the left-hand side is $0$ at the desired point(s), and you can easily check for which zeros of $f$ the derivative of the right-hand side is $0$.

share|improve this answer

Your Answer

 
discard

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.