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Unboundness and periodicity for complex trig functions

  1. Aug 13, 2013 #1

    ppy

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    Hi
    I just found out that cos(z) and sin(z) are unbounded and tend to ∞ which I find strange ! But the part i'm struggling with is that I can't reconcile that fact with the fact that they both have a period of 2pi. Surely that means that each value in the range 0-2pi is repeated in the range 2pi-4pi and so on ?
    Thanks
     
  2. jcsd
  3. Aug 13, 2013 #2
    Since we know that, for example, ##\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}##, if we set ##z=ix##, then ##\sin(ix)=\frac{e^{-x}-e^{x}}{2i}##.

    For values ##x_0\in(0,2\pi)##, is there a corresponding ##x_1\in(2\pi,4\pi)## for which ##\sin(ix_0)=\sin(ix_1)##? What does this imply about the periodicity of the sine function as the imaginary part of ##z## gets larger while the real part stays the same?
     
    Last edited: Aug 13, 2013
  4. Aug 13, 2013 #3

    ppy

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    I think I might be getting there ! The period is 2pi but there is no imaginary period ? So this means sin z repeats for every real value of 2pi but never repeats for imaginary values ?
     
  5. Aug 13, 2013 #4
    Essentially, yes.
     
  6. Aug 14, 2013 #5

    mathman

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    Science Advisor
    Gold Member

    For pure imaginary arguments, sin and cos are essentially sinh and cosh for real arguments.
     
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