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Finding the real and imaginary part

  1. Nov 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Determine the real part, the imaginary part, and the absolute magnitude of the following expressions:
    tanh(x-ipi/2)
    cos(pi/2-iy)


    2. Relevant equations
    cos(x) = e^ix+e^-ix
    tanh(x) = (1-e^-2x)/(1+e^-2x)

    3. The attempt at a solution
    for cos(pi/2-iy)= (e^(ipi/2-i^2(y))+e^(i^2(y)-ipi/2))/2
    =0.5(e^ipi/2*e^-i^2y + e^i^2y*e^-ipi/2)
    =0.5(ie^y+e^-y*-i)
    =0.5i(e^y-e^-y)
    therefore, imaginary = 0.5(e^y-e^-y)
    and real = 0

    for tanh(x-ipi/2) = (1-e^-2(x-ipi/2))/(1+e^-2(x-ipi/2))
    after simplifying i get
    (1+e^-2x)/(1-e^-2x)
    so that ^ is the real part
    and imaginary = 0

    I'm not really sure if im doing this right though, or if i have to somehow simplify these expressions to get the answer. If so, how would I solve the question?
     
  2. jcsd
  3. Nov 9, 2013 #2

    FeDeX_LaTeX

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    It's simpler to note that ##\cos\left(\frac{\pi}{2} - iy\right) = \sin(iy)##.

    There's a sign error here, otherwise, your final answers are correct -- it's a lot easier if you write it in terms of a hyperbolic trig function.

    That's correct too. Can you write that answer in terms of a hyperbolic trig function?
     
  4. Nov 9, 2013 #3
    i cant find the sign mistake that you're referring to. and yes, the second one would be coth(x)
     
  5. Nov 9, 2013 #4

    FeDeX_LaTeX

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    My apologies, no sign error. Looks correct to me.
     
  6. Nov 9, 2013 #5
    Alright thanks, as for finding the absolute magnitude, would the expressions above simply be the answer?
    if absolute mag = |a + bi| = sqrt(a^2 +b^2)= sqrt(0^2+0.5(e^y-e^-y)^2) =0.5(e^y-e^-y)
    for the second = sqrt(coth(x)^2) = coth(x)
     
  7. Nov 9, 2013 #6

    FeDeX_LaTeX

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    I'm not convinced about that. ##\sinh(x)## and ##\coth(x)## can take on negative values, can they not?
     
  8. Nov 9, 2013 #7
    So it's just going to be the + and - values of the above? Since you're taking the square root of it.
     
  9. Nov 9, 2013 #8

    FeDeX_LaTeX

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    Yes (for positive and negative arguments). Either way, ##|\coth(x)| = \coth(x)## and ##|\sinh(x)| = \sinh(x)## definitely do not hold true for all x! It's best to either leave it in absolute value form or define your function separately for different values of x.
     
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