MAT334--2020F > Quiz 4

Quiz4 Lec5101 5E

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Yuyan Liu:
Problem: Evaluate the given integral using the technique of Example 10 of Section 2.3:

$$\int_\gamma \frac{dz}{z^3} ,$$

where $\gamma$ is any curve in $\{z: Re z \geq 0, z \neq 0 \}$, joining $-i$ to $1 + i$.

Answer:

since f is analytic in all $Re(z) \geq 0, z \neq 0, \int_\gamma \frac{dz}{z^3} = F(1+i) - F(-i)$

$$F'(z) = f(z) \Rightarrow F(z) = - \frac{1}{2z^2}$$
$$F(1+i) = - \frac{1}{2(1+i)^2} = - \frac{1}{4i}$$
$$F(-i) = - \frac{1}{2(-i)^2} = \frac{1}{2}$$

Thus, $\int_\gamma \frac{dz}{z^3} = F(1+i) - F(-i) = - \frac{1}{4i} - \frac{1}{2} = \frac{2i-1}{4i}$

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