# Laurent expansion of principal root

1. May 2, 2009

### bernardbb

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

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

2. May 2, 2009

### Hurkyl

Staff Emeritus
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. May 2, 2009

### 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...

Last edited: May 2, 2009
4. May 20, 2011

### santoshrana21

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

5. May 20, 2011

### HallsofIvy

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