Laurent expansion of principal root

[bernardbb]

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

 

[Hurkyl]

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?)

 

[bernardbb]

Around any point a with |a|>1.

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

 

[santoshrana21]

is there any common and simple procedure to find the Laurent expansion of any function? If any then please reply me soon. Thanks

 

[HallsofIvy]

It's essential Taylor's series, allowing negative powers.