Laurent series question!

Well now it's below...

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I don't understand the comments. First, note that your expansion has a minus sign in front of 1/(z+1). So I had to add that in when I put the series into the final answers. Also, in both answers the term 1/z is not included in the argument of Σ. The answers in a) and b) are different from one anothe

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r. They should be though. The series from a) has an expanded form
1/z - 1 - 1/2 + z - z/4 - z² - z²/8 + ....

Some of those terms can obviously be combined.

The second series can be expanded as well.
... + 1/z^4 -1/z^3 + 1/z² - 1/2 - z/4 - z²/8 - ...

For the second one, the term 1/z doesn't

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appear. You have -1/z from 1/(1 + z) and 1/z which cancel.

Don't just look at the answers. Dissect the process. That is the important thing. Especially if you want to be able to do this yourself. You might find a typo in there, but I was careful when typing this up so I am not aware of any.

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Oh! I should add that Matt's second series is incorrect. His contribution from 1/(z - 2) is convergent for |z| > 2, but the instructions say 1 < |z| < 2. So don't compare his answer to mine.

I'm not knocking him. He clearly knows what he is doing. He just read the instructions incorrectly.

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I know. I get the process. I was just comparing the series when factored out (yours and Matts that is). The terms for the first series when 0 < |z| < 1 didn't match up either so I was wondering what was up with that. Does that make sense?

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(a) for |z|<1,

1/(z+1) = 1-z+z^2-...

1/(z-2) = 1/( 2(z/2-1) ) = -1/2( 1 - z/2 +(z/2)^2 - ... )

So, the Laurent expansion in this region would be

f(z) = 1/z - (1-z+z^2-...) -1/2( 1 - z/2 +(z/2)^2 - ... )
=1/z-3/2+5/4*z-9/8*z^2...

(b) For 1<|z|<2, the expansion of the denominators must be expanded on something small, such as 1/z, or 2/z.

1/(z+1) = 1/( z*(1+1/z) ) = (1/z)*(1+1/z)
=1/z*(1-1/z+(1/z)^2...)

1/(z-2) = 1/( z(1-2/z) ) = (1/z)*( 1+2/z + (2/z)^2... )
=1/z*(1+2/z+(2/z)^2...)


and so in this region,

f(z) = 1/z - (1/z - 1/z^2 +...) + ( 1/z + 2/z^2 +... )
=1/z + 3/z^2 ...