Author Topic: LEC0101 Quiz#4 oneA  (Read 314 times)

Xun Zheng

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LEC0101 Quiz#4 oneA
« 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
$$\int_{γ} \frac{dz}{z^2} = [-\frac{1}{z}]^{1+i}_{-i} = - \frac{1}{1+i}-\frac{1}{i}$$