Results 1 to 3 of 3

Thread: Asmptotic analysis problem

  1. #1
    Junior Member
    Joined
    May 2008
    Posts
    36

    Asmptotic analysis problem

    $\displaystyle f(z) \approx \sum_{n=0}^{\infty}{a_{n}z^{-n}} \ \ as \ \ z\rightarrow\infty$

    show by induction $\displaystyle \frac{1}{f(z)} \approx \frac{1}{a_{0}}\sum_{n=0}^{\infty}{d_{n}z^{-n}} \ \ where \ \ \sum_{k=0}^{n}{d_{n-k}a_{k}} $
    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
    A complex variable function $\displaystyle \varphi(s)$ analytic in $\displaystyle s=0$ can be written as...

    $\displaystyle \varphi(s)=\sum_{n=0}^{\infty} a_{n}\cdot s^{n}$ (1)

    If in (1) is $\displaystyle a_{0} \ne 0$ then $\displaystyle \frac{1}{\varphi(s)}$ is also analytic in $\displaystyle s=0$ and is...

    $\displaystyle \frac{1}{\varphi(s)}=\sum_{n=0}^{\infty} d_{n}\cdot s^{n}$ (2)

    A direct way to compute the $\displaystyle d_{n}$ from the $\displaystyle a_{n}$ is given by the identity we obtain combining (1) and (2)...

    $\displaystyle \varphi(s) \cdot \frac{1}{\varphi(s)} = \sum_{n=0}^{\infty} a_{n}\cdot s^{n} \cdot \sum_{n=0}^{\infty} d_{n}\cdot s^{n} = \sum_{n=0}^{\infty}s^{n}\cdot \sum_{k=0}^{n}a_{k}\cdot d_{n-k} = 1$ (3)

    ... so that is...

    $\displaystyle \sum_{k=0}^{n} a_{k}\cdot d_{n-k}= \left\{\begin{array}{cc}1, &\mbox {if } n=0\\0, & \mbox {if } n>0\end{array}\right. $ (4)

    If we perform the substitution of variable $\displaystyle s=\frac{1}{z}$ so that is...

    $\displaystyle \lim_{z \rightarrow \infty} \varphi (z)= \lim_{s \rightarrow 0} \varphi (s)$

    $\displaystyle \lim_{z \rightarrow \infty} \frac{1}{\varphi (z)}= \lim_{s \rightarrow 0} \frac{1}{\varphi (s)}$ (5)

    The conclusion is that if...

    $\displaystyle \lim_{s \rightarrow 0} \frac{\varphi(s)} {f(s)}=1$ (6)

    ... then it is also...

    $\displaystyle \lim_{s \rightarrow 0} \frac{f(s)}{\varphi(s)} =1$ (7)

    Kind regards

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

  3. #3
    Junior Member
    Joined
    May 2008
    Posts
    36

    idont

    i dont understand y u prove (7)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. analysis problem
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Oct 12th 2009, 08:47 AM
  2. analysis problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jan 5th 2009, 05:35 PM
  3. A problem in analysis
    Posted in the Calculus Forum
    Replies: 7
    Last Post: Jan 2nd 2009, 04:51 AM
  4. Analysis problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Oct 30th 2008, 01:02 PM
  5. please help, analysis problem..
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Aug 28th 2008, 05:03 AM

Search Tags


/mathhelpforum @mathhelpforum