Page-99 , VCA by Tristan Needham
In this page, it is shown how as we circular about the circle $\omega$ moves to different points. I am struggling to understand how the curve bends when moving. Could someone explain that?
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Disintegrating By Parts
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user123234
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Does this picture of the Riemann surface of log (taken from here) help? – Andrew D. Hwang May 31 '22 at 16:16
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You should add the paragraph that you got confused so that other people can undersstand the context of what you're asking. @Wave – tryst with freedom May 31 '22 at 17:39
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In your question, you seem to be taking two branches. I tried checking that section and he works it out for only one branch to my understanding. Are you referencing multipile pages or only the one which pic comes in – tryst with freedom May 31 '22 at 17:55
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@Aplateofmomos I added the paragraph. Why do you think is he only taking one branch I mean if he add $+2\pi i$ doesn't need to take another branch? – user123234 May 31 '22 at 18:08
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Ok, so you are saying the function switches over to two different branches depending on motion? – tryst with freedom May 31 '22 at 18:09
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@Aplateofmomos Hmm I don't really understand your rephrased scentense because of my english but i think so. I mean if we takes the natural branch of logarithm to compute $\log(z)$ then why did he add $2\pi i$? I thought that he looks at $\log(z)$ in the "next" branch. – user123234 May 31 '22 at 18:12
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I mean for comparing such branches you need to actually have a branch cut. Till the point you have said, he hasn't even made a cut yet to my understanding @Wave – tryst with freedom May 31 '22 at 18:15
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@Aplateofmomos but what is a branchcut? Maybe that's also a missunderstanding. – user123234 May 31 '22 at 18:17
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I have written about it a bit in my answer. Maybe that will help. Let me know if it does @Wave – tryst with freedom May 31 '22 at 18:17
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Let us continue this discussion in chat. – user123234 May 31 '22 at 18:21
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@AndrewD.Hwang so is your picture the picture of the logarithm function viewed as multivalued function? – user123234 May 31 '22 at 19:26
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1In a word, yes. Precisely, it's the sum of the real and imaginary parts, which shows both the undefinedness at $0$ and the multi-valued nature. – Andrew D. Hwang May 31 '22 at 19:33
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@AndrewD.Hwang Perfect thanks! – user123234 May 31 '22 at 19:37
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I won't try and give some calculation with number but rather a qualitative answer for you. I'll show how to "see" the situation qualitatively for the counter-clockwise motion.
Firstly notice that if you were to move around the circle counter-clockwise, initially the magnitude of $|z|$ is decreasing as the point on circle is closer and closer to origin and later after the redpoint I marked here, it will increase again:
What about the angle? Well if you were initially at $\theta$, as you spin around the circle, you gain more and more angle till you return to the initial point where you have total $2\pi$ more angle. That why the curve is going up.
tryst with freedom
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Could you say that differently? I don't think I understood you @Wave – tryst with freedom May 31 '22 at 18:16
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I mean if this continuous black curve in the right picture turns around the Y-axis like when you wrap a cord around a stick. – user123234 May 31 '22 at 18:18
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I mean if you do that in a real life stick there is a 3 dimensional aspect going on, here it is 2d, but it does look like that if that's what you are saying @Wave – tryst with freedom May 31 '22 at 18:19
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