Complex analysis: Singularities

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This lecture is part of an online undergraduate course on complex analysis.

We discuss the different sorts of singularities of a holomorphic function (removable singularities, poles, essential singularities, branch-points, limits of singularities, natural boundaries) and give examples of each type.

In the comments Romain Gicquaud pointed out an easier proof of the removable singularity theorem: just consider z^2 f near 0, and observe that it is differentiable.

  1. Richard E Borcherds
  2. Introduction
  3. Complex analysis
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Комментарии
Автор

the only person speaking english on this topic on youtube!!

surrendereverything
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"you are being deliberately perverse" is my initial reaction to the Weierstrass function.

Someone-crcj
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For the removable singularity result, it is easier to consider g(z) = z^2 f(z) which is differentiable everywhere including at the origin and has continuous derivative so g is holomorphic. As it has power expansion starting at z^2 we conclude that f(z) = g(z) /z^2 is holomorphic everywhere. Nice video as always !

grog-im
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What a great video! I burst out laughing hearing someone call an essencial singularity "nasty"

poliana
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6:22 "when a fuction is BOUNDED". woke me up XD

albertgathercole
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Thanks Prof. Borcherds, this was a eally enjoyable lecture. Finally we're moving beyond what I remember from my complex analysis course in 1998-99.

Could we see a proof of Picard's theorem? Or maybe a reference to a readable version of it?

And towards the end of this course, could you give a quick survey of "several complex variables" in comparison/contrast to "1 complex variable".

nonindividual
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Thanks, this was very well done and explained.

AdrienLegendre
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Many of the figures in this video came from a book

Is that book just called table of functions by jankhe and edme?

I can't exactly tell what the title was but this book came up after a search and I'm wondering if someone could verify

fanalysis
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Great playlist, what book is this you're using for diagrams? Thanks!

peterpaton
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'They've only drawn a finite number of poles.' Haha. Well that's just not good enough is it?

Vidrinskas