Maclaurin and Laurent series in the complex plane.

**alucard2006** · April 18th 2010, 04:03 PM

Let f(z) = (z+2)/(z-2)

Find the Maclaurin series for f on the domain abs(z) < 2. Find the Laurant series for f centered at z0 = 0 on the domain 2 < abs(z) < Inf. And find the Laurant Series for f centered at z0 = 2 on the domain 0 < abs(z-2) < Inf.

**chisigma** · April 18th 2010, 08:14 PM

Is...

$f(z)= \frac{z+2}{z-2} = 1 + \frac{4}{z-2} = 1 - \frac{2}{1-\frac{z}{2}} =$

$= 1 - 2\cdot \sum_{n=0}^{\infty} (\frac{z}{2})^{n} = -1 - z - \frac{z^{2}}{2} - \frac{z^{3}}{4} - \dots$ (1)

The (1) converges for $|z|<2$ . Setting in (1) $s=\frac{1}{z}$ we have...

$f(s)= \frac{1 + 2s}{1-2s} = -1 - \frac{1}{s} - \frac{1}{2 s^{2}} - \frac{1}{4 s^{3}} - \dots$ (2)

... and the (2) converges for $|s|>2$ ...

Kind regards

$\chi$ $\sigma$

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