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Hyperbolic functions problem

  1. May 5, 2009 #1
    Given the quantity

    Cosh(x)*Cosh(y)


    where x and y are two indipendent real variables is it possible to write it only in function of


    k=Cosech(x)*Cosech(y)

    ?????
    It could seem a quite easy problem but I spent a few days between the proprieties of hyperbolic functions and I really didn't find a way to solve it.
     
  2. jcsd
  3. May 7, 2009 #2
    Hint:
    Use the identity (cosh(t))^2-(sinh(t))^2=1 to solve for cosh(t).

    Then use the fact that csch(t) = 1/sinh(t) so sinh(t) = 1/csch(t).
     
  4. May 7, 2009 #3
    thank you for the hint Russell,
    but that's not a solution to my problem, as I want to write the quantity

    Cosh(x)*Cosh(y)


    ONLY in function of k. If I did like you suggested me, I find terms like


    Sinh(x)+Sinh(y)


    and I can't find a way to write them in function of k.
     
  5. May 7, 2009 #4
    It is not possible.
    Assume you have some function f(k) that represents cosh(x)cosh(y) in terms of k.
    When k = 1/2, then what would f(k) be?

    Let sinh(a)=.5, sinh(b)=4. Then k = 1/(.5*4) = 1/2
    Then cosh(a)*cosh(b)=sqr( 1+1/4)*sqr( 1+16)=sqr(85/4)=f(1/2)


    However look at:
    let sinh(c)=1, sinh(d)=2. Then k = 1/(1*2) = 1/2
    But cosh(a)*cosh(b)=sqr( 1+1)*sqr( 1+4)=sqr(10)=f(1/2)

    So, f(1/2) would not have a single output value, it is not a function.
     
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