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Give me a concrete example, please.

Now, let \(z =\sqrt2 /2 + i\sqrt 2/2 \) hence arg z = pi/4,
How can I define alpha?

Should I take alpha = pi/4 also?
hence the region is \(C - \{re^{i\pi/4}\} , r\geq 0\)

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what do you mean take \(z=e^{i\pi/4}\), you need to pick a range of angles to use for the plane

I didn't say that. I said if \(z = \sqrt 2/2 + i\sqrt 2/2\) then \(arg z = \pi/4\)

Is there any video clip I can study ? I want to completely understand what it is. Please.