Expanding in powers of 1/z (Laurent series)

  1. The text book used in one of my courses talks about expanding functions in powers of 1/z aka negative powers of z.

    The problem is that I cannot recall that any previous course taught me/challenged me on how to expand functions in negative powers. For example, Taylor series only have positive powers.

    Is there a general method of expanding in negative powers, like for Taylor series, or are there at best similar methods for similar functions?

    I fear I have overlooked something elementary here, because I feel strangely clueless about this one (and Internet searches have made me no wiser). The textbook only gives examples of the results of expansion in 1/z, but never gives any details on how it is done.
     
  2. jcsd
  3. Office_Shredder

    Office_Shredder 4,498
    Staff Emeritus
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    Is this a complex analysis course, or is it something that had complex analysis as a prerequisite?

    Practically, to calculate these you can often do standard Taylor series calculations
    [tex]f(x)=\frac{x}{1-x} = \frac{1}{1-1/x}[/tex]

    we know how to expand 1/(1-1/x) using the Taylor series for 1/(1-x)
    [tex] f(x) = 1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+...[/tex]
    and this is valid as long as |x|>1
     
  4. It's a physics course without physics, if that makes any sense. To answer the question though, complex analysis is a part of the course rather then a prerequisite (e.g. it includes the most basic proofs/definitions for differention of functions of a complex variable).

    I do recall seeing the Taylor expansion you introduced (in an introductory course in astropysics, as a matter of fact).

    Anyway, I guess my question has been answered.
     
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