# Estimate ∫γ dz/(1 + z^4) as R→∞.

1. Sep 8, 2011

### blueyellow

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?
IT never says that $iRe^{it}=R$. What you use is that the modulus of that is R. Thus
$$|iRe^{it}|=|i||R||e^{it}|=1$$