Help me with a Laurent series
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Help me with a Laurent series

[From: ] [author: ] [Date: 11-12-09] [Hit: ]
..Can I manipulate 1 / (z + z²/2! + z³/3! + ........
I need to find the principal part of the Laurent series of the following function:

1/(e^z - 1)

I found the power series for (e^z - 1) = z + z²/2! + z³/3! + ...

Can I manipulate 1 / (z + z²/2! + z³/3! + ...) to figure out the principal part (negative values) of the Laurent series? I'm clearly failing at algebra tonight. Any help is appreciated!

-
Due to z being the term of smallest positive degree, we see that the principal part of 1/(e^z - 1) is 1/z.

Why?

If 1/(e^z - 1) = 1/z + A + Bx + ..., then
(e^z - 1) (1/z + A + Bx + ...) = 1
==> (1/z + A + Bx + ...)(z + z²/2! + z³/3! + ...) = 1
==> 1 + (constant term and terms of higher degree) = 1, via distributive law.

(If needed, we see that the constant term and beyond must equal 0; so we could solve for A, B, etc. as seen fit.)

I hope this helps!
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