Complex analysis: Summing series

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This lecture is part of an online undergraduate course on complex analysis.
This is a replacement for a previous video, correcting some minor typos.

We show how to use the residue calculus to sum series, such as Euler's series 1/1^2 + 1/2^2+ ...

Solution to exercise in rot 13: cv phorq bire guvegl gjb

  1. Richard E Borcherds
  2. Introduction
  3. Complex analysis
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Комментарии
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That was amazing! Really shows the beauty of this subject!
In the last exercise I got that the sum is π³/32, did anybody got the value too?

leeholzer
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At 2:45 I don't get how the residues of 1/(z^2 tan z) are determined: Prof Borcherds says that at -2pi and 2pi the residues are 1/(2pi)^2, and at -pi or +pi they are 1/pi^2. On the other hand, later at 9:00, he shows from the expansion of 1/(z^2 tan z) = (1/z^2) cot(z) = (1/z^2) (1/z - z/3 ...) = 1/z^3 - 1/3z ..., that the residue is -1/3, which makes sense. But there seems to be a discrepancy.

reinerwilhelms-tricarico
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Was this video re-uploaded? I'm sure I saw it about a week ago.

peterboneg
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In the description, what's the solution to the exercise?

VaslavTchitcherine
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This is an introductory course ... I will show a straightforward application of this difficult concept just introduced ... proceeds to a problem where subtle artifices are absolutely required ... same issue in video 12

a gift horse and all that ... but smh

dacianbonta