Results 1 to 2 of 2

Thread: isolated point, Cauchy integral formula

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    23

    isolated point, Cauchy integral formula

    Let $\displaystyle z_0$ be an isolated point of a function $\displaystyle f$ and suppose that $\displaystyle f(z)=\frac{\phi(z)}{(z-z_0)^m}$, where $\displaystyle m$ is a positive integer and $\displaystyle \phi(z)$ is analytic and nonzero at $\displaystyle z_0$. By applying the extended form of the Cauchy integral formula to the function $\displaystyle \phi(z)$, show that $\displaystyle \text{Res}_{z=z_0}=\frac{\phi^{(m-1)}(z_0)}{(m-1)!}$.

    I do not see how to do this. Our book says:
    Since there is a neighborhood $\displaystyle |z-z_0|< \epsilon$ throughout which $\displaystyle \phi(z)$ is analytic, the contour used in the extended Cauchy integral formula can be the positively oriented circle $\displaystyle |z-z_0|< \frac{\epsilon}{2}$. How do I use this suggestion? I don't see that now. Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    6
    The defintion of 'residue' of an $\displaystyle f(z)$ which has a pole of order m in $\displaystyle z=z_{0}$ and is analytic elsewhere in a region around $\displaystyle z_{0}$ is ...

    $\displaystyle Res_{z=z_{0}} f(z) = \lim_{z \rightarrow z_{0}} \frac{1}{(m-1)!} \frac{d^{m-1}}{dz^{m-1}} \{(z-z_{0})^{m}\cdot f(z)\}$ (1)

    Now if is...

    $\displaystyle f(z)= \frac{\phi(z)}{(z-z_{0})^{m}}$ (2)

    ... where $\displaystyle \phi(*)$ is analytic in $\displaystyle z=z_{0}$ the (1) gives to You...

    $\displaystyle Res_{z=z_{0}} f(z) = \frac{1}{(m-1)!}\frac{d^{m-1}}{dz^{m-1}} \phi(z)_{z=z_{0}} $ (3)

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cauchy Integral Formula
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Jun 16th 2011, 08:40 AM
  2. Cauchy's integral formula help
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: May 10th 2010, 05:31 PM
  3. Cauchy Integral Formula
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Nov 1st 2009, 11:55 AM
  4. [SOLVED] Line integral, Cauchy's integral formula
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Sep 16th 2009, 11:50 AM
  5. [SOLVED] Line integral, Cauchy's integral formula
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Sep 15th 2009, 01:28 PM

Search Tags


/mathhelpforum @mathhelpforum