Laurent expansion of principal root

In summary, to find the Laurent expansion of a function containing the principal branch cut of the nth root, you would follow the same procedure as for any other function. This can be done around any point a with |a|>1. The principal nth root of z can be defined as \sqrt[n]{|z|}e^{\frac{1}{n}i\mathrm{Arg}z}, but this may not be very helpful for finding an expansion. It may be necessary to use Taylor's series and allow for negative powers. There is no common and simple procedure for finding the Laurent expansion of any function.
  • #1
bernardbb
4
0
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]
 
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  • #2
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
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
It's essential Taylor's series, allowing negative powers.
 

What is the Laurent expansion of principal root?

The Laurent expansion of a principal root is a mathematical series that represents a complex function in terms of its principal root. It allows for the approximation of the function near its singular point.

What is the difference between the Laurent expansion of principal root and Taylor series?

While both the Laurent expansion of principal root and the Taylor series are mathematical series used for approximation, the main difference is that the Laurent expansion includes negative powers of the variable, while the Taylor series only includes non-negative powers.

What is the significance of the principal root in the Laurent expansion?

The principal root is the point around which the function is being approximated. It is typically a singular point, such as a pole or branch point, where the function is undefined or non-analytic. The Laurent expansion allows for the calculation of the function's behavior near this point.

Can the Laurent expansion of principal root be used for all functions?

No, the Laurent expansion can only be used for functions that are analytic in a region around the principal root. This means that the function must have a continuous derivative of all orders in this region.

How is the Laurent expansion of principal root calculated?

The Laurent expansion is calculated using a combination of the function's Taylor series and its Laurent series, which includes negative powers of the variable. The coefficients of these series are determined through a process of differentiation and integration.

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