Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I A question about the log of a rational function

  1. Aug 7, 2016 #1
    We have the rational function :
    $$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\left(\frac{1-ix}{1+ix}\right)^{n/2}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$
    It's not hard to prove that :
    $$\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}=(-1)^{n}\prod_{k=1}^{n-1}\frac{x+i(\xi_{n}^{k}-1)}{x-i(\xi_{n}^{k}-1)}\;\;\;,\;\;\xi_{n}^{k}=e^{2\pi i k/n}$$
    Now we want to compute $$\log f(x)$$ for x>0. The logarithm of the individual factors can be written as :

    $$\log\left(\frac{x+i(\xi_{n}^{k}-1)}{x-i(\xi_{n}^{k}-1)}\right)=2i\tan^{-1}\left(\frac{x}{1-\xi_{n}^{k}}\right)+i\pi;\;\;\;\;x>0$$
    So, one would expect:
    $$\log f(x)=-in\tan^{-1}(x)-i\pi+2i\pi n+2i\sum_{k=1}^{n-1}\tan^{-1}\left(\frac{x}{1-\xi_{n}^{k}}\right)$$
    But it looks nothing like what wolframalpha returns. What am i doing wrong here ?
     
    Last edited: Aug 7, 2016
  2. jcsd
  3. Aug 7, 2016 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    It might be helpful to say what wolfram alpha returns.
     
  4. Aug 7, 2016 #3
    graphically, there seems to be a difference between what i've calculated and the plot of ##\log f(x)## by multiples of ##2\pi## . but i am not able to locate the exact locations of the jumps.
     
  5. Aug 7, 2016 #4

    fresh_42

    Staff: Mentor

    You spent so much effort to type in post #1 and then you fail with some copies and pastes or links?
    I hope micromass' crystal ball isn't out of order like mine currently is.
     
  6. Aug 7, 2016 #5
    have you ever used WF ? it doesn't return results for general n !! and posting one example won't be of help if it doesn't say where the jumps are ! thanks for the very helpful and constructive post anyways !!
     
  7. Aug 7, 2016 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    If you refuse to give further information, then there is not much we can do. All I can say is that the complex logarithm is multivalued, so this can explain jumps of order ##2\pi##.
     
  8. Aug 7, 2016 #7
    where should i expect the jumps to happen ? that's where i am stuck. and we can just forget about the graphical discrepancy and correct my analytic calculation.
     
  9. Aug 7, 2016 #8

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Why don't you show us the graphics you got? I won't reply further to this thread if you don't show us what you did in wolframalpha.
     
  10. Aug 7, 2016 #9
  11. Aug 7, 2016 #10
    it boils down to finding the discontinuities of ##\log f(x)##
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: A question about the log of a rational function
  1. Rational functions (Replies: 3)

  2. Rational functions (Replies: 2)

Loading...