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Estimate ∫γ dz/(1 + z^4) as R→∞.

  1. Sep 8, 2011 #1
    1. The problem statement, all variables and given/known data

    Estimate ∫γ dz/(1 + z^4) as R→∞.

    Note that letting z = Re^(it) for t in [0, π]:
    |∫γ dz/(1 + z^4)|
    = |∫(t = 0 to π) (iRe^(it) dt) / (1 + R^4 e^(4it))|
    ≤ ∫(t = 0 to π) R dt / |1 + R^4 e^(4it)|
    ≤ ∫(t = 0 to π) R dt / (R^4 - 1), since R > 1
    ≤ πR / (R^4 - 1).

    but why does iRexp(it)=R ?
    why does i exp(it)=1?
    please help
     
  2. jcsd
  3. Sep 8, 2011 #2

    micromass

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    IT never says that [itex]iRe^{it}=R[/itex]. What you use is that the modulus of that is R. Thus

    [tex]|iRe^{it}|=|i||R||e^{it}|=1[/tex]
     
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