Asmptotic analysis problem

**pengchao1024** · October 25th 2009, 10:39 PM

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

show by induction $\frac{1}{f(z)} \approx \frac{1}{a_{0}}\sum_{n=0}^{\infty}{d_{n}z^{-n}} \ \ where \ \ \sum_{k=0}^{n}{d_{n-k}a_{k}}$

**chisigma** · October 26th 2009, 01:40 AM

A complex variable function $\varphi(s)$ analytic in $s=0$ can be written as...

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

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

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

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

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

... so that is...

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

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

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

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

The conclusion is that if...

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

... then it is also...

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

Kind regards

$\chi$ $\sigma$

**pengchao1024** · October 26th 2009, 09:26 AM

i dont understand y u prove (7)

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