Saturday, 15 June 2013

Evaluate intinftyinftyfracdx1+x2 using complex integration




I'm trying to evaluate the real integral dx1+x2



Denote Γ1=[R,R] Γ2=Reit, for t[0,π],
and let γ be a small circle around i so γ is inside
the area bounded by Γ1Γ2. By Cauchy's theorem:
Γ1f(z)dz+Γ2f(z)dz=γf(z)dz
And calculating γf(z)dz gives us π

(operating Cauchy's formula on the function 1z+i). so
we got
Γ1f(z)dz+Γ2f(z)dz=π
now I need to show that
lim
and I'm stuck.


Answer



You can apply Estimation lemma. Since
\left|\int_{\Gamma_2} \frac{1}{1+z^2}dz\right| \le \frac{\pi R}{R^2 -1}
for large R,
\lim_{R\to\infty}\left|\int_{\Gamma_2} \frac{1}{1+z^2}dz\right|=0.
Then you can get what you want.


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