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2) Determine and sketch the analycity region for log($z^2$), then compute it's derivative.

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5) Evaluate the contour integral where D is the upper semicircle centered at 0 with radius 2 from 2 to -2, $\int_D \frac{1}{z^3}dz$.

Take $t \in [0, \pi]$. Let $w(t) = 2e^{it}$, so $w'(t) = 2ie^{it}$. Then

$\int_2^{-2}\frac{1}{(2e^{it})^3} 2ie^{it}dt = i\int_2^{-2}\frac{1}{(2e^{it})^2}dt = i\int_2^{-2}(2e^{it})^{-2}dt = [i\frac{ie^{2it}}{8}]_2^{-2} = \frac{e^{-2i(-2)}}{8} - \frac{e^{-2i(2)}}{8}$

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6) Evaluate $\int_D f dz$, where $f(x+iy) = x^2 + iy^2$ and $D$ is the line joining $1$ to $i$.

$\int_1^0 x^2dx - \int_0^i y^2 dy + i[\int_1^0 y^2 dx + \int_0^i x^2 dx]$ - is this on the right track?

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7): Evaluate $\int_0^{2\pi}e^{e^{i\theta}}e^{i\theta}d\theta$.

$\int_0^{2\pi} e^{e^{i\theta}} e^{i \theta} d\theta = [-ie^{e^{i\theta}}]_0^{2\pi} = 0$. Alternatively, because f(z) is entire, the integral is equal to 0.

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8) Evaluate $\int_D \frac{1}{z^2 - 4z} dz$ where $D$ is the counterclockwise circle of radius 3 centered at 4.

Tue, 09 Nov 2021 00:20 GMT