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Math Help - easy proof complex analysis

  1. #1
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    easy proof complex analysis

    Prove that Log e^z = z if and only if -pi<Im<pi


    can i have a little help with this?
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  2. #2
    Junior Member
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    Log(exp(z))=ln(|exp(z)|)+iarg(exp(z))=x+iarg(exp(z ))
    z=x+iy
    Now, e^z=exp(x)(cos(y)+isin(y))
    So arg(e^z)=y iff y is between pi and minus pi or between 0 and 2pi, as long as it doesn't passes a length of 2pi, because then an increment of 2pi is added to the argument.
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