### Author Topic: Q5 TUT 0203  (Read 2740 times)

#### Victor Ivrii

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##### Q5 TUT 0203
« on: November 02, 2018, 03:30:59 PM »
$\newcommand{\Log}{\operatorname{Log}}$
Find the first four terms in power-series expansion about the given point for the given function; find the largest disc in which the series is valid:
$$[\Log (1-z)]^2\qquad\text{about}\; z_0 = 0.$$
« Last Edit: November 04, 2018, 09:13:14 PM by Victor Ivrii »

#### Meng Wu

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##### Re: Q5 TUT 0203
« Reply #1 on: November 02, 2018, 03:59:37 PM »
We know \begin{align}Log(1-z)&=-\sum_{n=1}^{\infty}\frac{z^n}{n}=-(z+\frac{z^2}{2}+\frac{z^3}{3}+\cdots)\\ \Rightarrow [Log(1-z)]^2&=[-(z+\frac{z^2}{2}+\frac{z^3}{3}+\cdots)]^2 \\&=(z+\frac{z^2}{2}+\frac{z^3}{3}+\cdots)^2 \\ &=(z+\frac{z^2}{2}+\frac{z^3}{3}+\cdots)(z+\frac{z^2}{2}+\frac{z^3}{3}+\cdots)\\ &=0+(0)z+(1)z^2+(1)z^3+\frac{11}{12}z^4+\frac{5}{6}z^5+\cdots\end{align}
Series is valid at $|z|<1$.
« Last Edit: November 02, 2018, 04:21:15 PM by Meng Wu »