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Length of a contour (easy)

  1. Apr 22, 2006 #1
    Find the length of the contour Gamma parametrized by [itex] z = z(t) = 5e^{3it} [/itex]

    the legnth of contour is [tex] \int_{0}^{\pi} \left|\frac{dz}{dt}\right| dt [/tex]

    now [tex] \left|\frac{dz}{dt}\right| = |15ie^{3it}| = |15ie^{3t}\cos(t) - 15e^{3t}\sin(t)| = \sqrt{225e^{6t} + 225e^{6t}} [/tex]

    is this right so far?? I think im making a mess of the magnitude part
     
  2. jcsd
  3. Apr 22, 2006 #2

    nrqed

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    The magnitude of [itex]15ie^{3it}[/itex] is 15!! (just take the expression times its complex conjugate and then take the square root!).

    Your second step is all wrong. You seem to think that [itex] e^{it} = e^t e^i[/itex] ?? That`s of course incorrect!

    Euler`s identity applied to [itex] e^{3 i t} [/itex]gives [itex] cos(3t) + i sin(3t)[/itex]

    EDIT: which can also be written as [itex] (e^{it})^3 = (cos t + i sin t)^3[/itex] of course
     
    Last edited: Apr 22, 2006
  4. Apr 23, 2006 #3
    dont i feel stupid!
    i was using Euler's identitiy incorrectly, when my course... for the most part... is based on it!
     
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