Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Quiz 4 => Topic started by: Xun Zheng on October 23, 2020, 11:27:37 AM

Evaluate the given integral using the technique of Example 10 of Section 2.3:
$$\int_{γ} \frac{dz}{z^2}$$
where γ is any curve in {z: Re(z)≥0, z≠0}, joining i to 1+i.
Here is my answer:
First, we observe that γ is not closed.
Since γ is in {z: Re(z)≥0, z≠0}, then
$$f(z)=\frac{1}{z^2}$$ is analytic on D.
Thus we have
$$\int_{γ} \frac{dz}{z^2} = [\frac{1}{z}]^{1+i}_{i} =  \frac{1}{1+i}\frac{1}{i}$$