Laurent expansion of principal root

  • Thread starter bernardbb
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  • #1
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Main Question or Discussion Point

How do I find the Laurent expansion of a function containing the principal branch cut of the nth root?

Example:
[tex]f(z)=-iz\cdot\sqrt[4]{1-\frac{1}{z^{4}}}[/tex]
 

Answers and Replies

  • #2
Hurkyl
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In exactly the same way you would any other function. What's giving you trouble?

(p.s. about what point are you trying to find an expansion?)
 
  • #3
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Around any point a with |a|>1.

As far as I know, the principal nth root of z is defined as [tex]\sqrt[n]{|z|}e^{\frac{1}{n}i\mathrm{Arg}z}[/tex], which doesn't seem very helpful; how would you expand [tex]\sqrt{z}[/tex] around a point where it is defined?

Perhaps these are silly questions; my book is very vague on Laurent expansions...

EDIT: I think I've got it... or maybe not...
 
Last edited:
  • #4
is there any common and simple procedure to find the Laurent expansion of any function? If any then please reply me soon. Thanks
 
  • #5
HallsofIvy
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It's essential Taylor's series, allowing negative powers.
 

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