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Sinh series manipulation

  1. Apr 17, 2016 #1
    1. The problem statement, all variables and given/known data
    Hello, I'm not trying to solve this exact problem although mine is similar and I am confused on how they were able to get a -1 in the exponent from one step to another.

    2. Relevant equations
    I have attached a picture indicating the step that I am confused about. How are they able to manipulate the series and pull out that -1 into the exponent thereby finding the residue?

    3. The attempt at a solution
    Some sort of series manipulation that I can't figure out; any help is greatly appreciated, thank you guys!
     

    Attached Files:

  2. jcsd
  3. Apr 17, 2016 #2
    It comes from the reciprocal rule for exponents.

    In general, ## \frac{1}{x^n} = x^{-n} ##. So as an example:

    ## \frac{1}{x^4-x^6} = \frac{1}{x^4(1-x^2)} = x^{-4}(1-x^2)^{-1}##
     
  4. Apr 17, 2016 #3

    SteamKing

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    The common factor -(z - πi)3 can be factored out of the series expression of sinh3 z before it is inverted. The series involves only alternating odd powers of the common factor. After that, for the inversion, one can apply mfiq's hint about using the law of exponents.

    There's no magic here - just straightforward algebra.
     
  5. Apr 17, 2016 #4
    Awesome, makes perfect sense. Thank you both.
     
  6. Apr 17, 2016 #5
    If either of you can still see this, would you possibly be able to tell me how they then lose that -1 power on the next line down allowing them to find the residue? Thanks again.
     
  7. Apr 17, 2016 #6
    Nevermind, think I've got it now.
     
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