Here is the exercise:(adsbygoogle = window.adsbygoogle || []).push({});

Use the indefinite integral to compute [tex] \int_{C} \sqrt{z}dz[/tex] where C is a path from z = i to z = -1 and lying in the third quadrant. Note: [tex] \sqrt{z} = e^{(1/2)lnz}[/tex] where the principal branch of lnz is defined on [tex]C \setminus [0,\infty][/tex].

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I am just a little unsure of why he gave us that note (although I do use it). Here is what I did for the exercise:

[tex] \int_{C} \sqrt{z}dz = \left[ \frac{2}{3}z^{3/2} \right]_{i}^{-1} [/tex]

[tex]= \frac{2}{3} \left[ e^{(3/2)ln(-1)} - e^{(3/2)ln(i)} \right] [/tex]

[tex]= \frac{2}{3} \left[ e^{(3/2)\pi i} - e^{(3/2)(\pi/2) i} \right] [/tex]

[tex]= \frac{2}{3}(1 - 2i)[/tex]

Everything look fine?

Thanks.

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# Complex Integral

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