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Complex Hyperbolic Derivative Problem

  1. Sep 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Hello,
    Thanks for taking some time to help me out...and I have to apologize for posting a graphic of my logic and attempted answer instead of using LateX (It would take me a very long time just to get this problem viewable)

    Please help me check my work and logic here...(see attached graphic) I dont know of any place I can see if my answer is correct (Not a book problem)

    The problem itself is stated in the first box drawn (see below)...


    2. Relevant equations
    None


    3. The attempt at a solution

    My logic (with comments throughout) and the last red box drawn is my attempt at a solution..(see the graphic below)
    I hope this is sort of easy to follow ...and I appreciate any help.
    Thank you...

    l_144c147752ad415e86cbcb6f370a0fea.jpg

    Does this seem right?
     
  2. jcsd
  3. Sep 7, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    No, it doesn't seem right. At one point you have
    [tex]\frac{1}{2}\frac{1}{2}sech^2(\frac{x}{2})- \frac{1}{2}tanh(\frac{x}{2})sech(\frac{x}{2})[/tex] and then, in the next two lines, that difference has metamorphised into a product,
    [tex]\left(\frac{1}{4}\right)\left(\frac{1}{2}\right)\left(sech^2(\frac{x}{2})\right)\left(-1\right)\left(tanh^2(\frac{x}{2})\right)\left(sech^2(\frac{x}{2})\right)[/tex]
    and then in the next line, you have reassembled that difference into a sum!

    The first line I mention above,
    [tex]\frac{1}{2}\frac{1}{2}sech^2(\frac{x}{2})- \frac{1}{2}tanh(\frac{x}{2})sech(\frac{x}{2})[/tex]
    is correct. You can factor (1/2)sech2(x/2) out:
    [tex]\frac{1}{2}sech^2(x/2)\left(\frac{1}{2}- tanh^2(\frac{1}{2})\right)[/tex]
    but that's about all you can do.
     
  4. Sep 7, 2009 #3
    @HallOfIvy..

    Thanks very much for reviewing my work!

    I was very suspicious of that operation (I thought it might be illegal)...thats why I was hoping an expert such as yourself could give me some insight.

    Now Ive got two questions:

    1) The sbove mentioned operstion is illegal why?
    What I was trying to do was turn the difference into a sum by factoring out the negative -1?
    In effect turning function-function*function into (function+function)-1*function...
    Or is this totally illegal...

    2) So the final answer is:
    or should it be:
    [tex]\frac{1}{2}sech^2(x/2)\left(\frac{1}{2}- tanh^2(\frac{x}{2})\right)[/tex]
    ...and so one of these is then the FINAL answer?
     
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