Question: Let $r$ be a real number, $r > 0$ and let $Lr$ be the line from the point $−r$ to $r$ in $C$. Let $γr$ be the upper half circle with radius $r$ and center in $0$. $$ \ f(z) = \frac{1}{1+z^2}\,. $$
Compute $\int f(z)dz$ when $r>1$
Answer:
I having trouble answering the last question about when $r>1$, I have done line integral with $Lr$ and separately for $γr$ when $r<1$ using Cauchy's Integral Formula. Now for $r>1$
$$ \ \int_{γr}^\ f(z)\,dz $$ Using the Residue formula for the simple pole in the interior $z=\mathbb i$ I know for the contour integral where $γ$ is the join of $Lr$ and $γr$ that $$ \ \int_{γ}^\ f(z)\,dz = π $$ My problem is how to finish the answer. By splitting the last integral $$ \ \int_{γ}^\ f(z)\,dz = \int_{Lr}^\ f(z)\,dz + \int_{γr}^\ f(z)\,dz = π + 0 = π $$ I see that second integral on r.h.s goes to zero as $r$ grows to infinity. Is this the right way to approach? I have seen other examples and questions approaching exactly this, but where do I stop?