Say we have the function:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{1}{\left( z-1 \right)\left( z+2 \right)}[/tex]

Using partial fractions,

[tex]\frac{1}{\left( z-1 \right)\left( z+2 \right)}\; =\; \frac{1}{z-2}\; -\; \frac{1}{z\; -\; 1}[/tex]

My question comes in on why and how these equations are manipulted for different regions.

Now for a) region |z| < 1

[tex] \frac{1}{z-1}\; =\; -\frac{1}{1-z}\; =\; -\sum_{j=0}^{\infty }{z^{j}}\; [/tex]

But for region 1 < |z| < 2

[tex]\frac{1}{z-1}\; =\; \frac{1}{z}\frac{1}{1-\frac{1}{z}}\; =\; -\frac{1}{z}\sum_{j=0}^{\infty }{\frac{1}{z^{j}}\; =\; }\sum_{j=0}^{\infty }{\frac{1}{z^{j+1}}}[/tex]

I have no idea how or why they are being manipulated for different regions. My book assumes me to be brilliant I suppose?

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# Question about Laurent Series

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