Complex Analysis, Math 87012
[ garrett@math.umn.edu ]
( See also:
[ vignettes ]
...
[ functional analysis ]
...
[ intro to modular forms ]
...
[ representation theory ]
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[ Lie theory, symmetric spaces ]
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[ buildings notes ]
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[ number theory ]
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[ algebra ]
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[ complex analysis ]
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[ real analysis ]
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[ homological algebra ]
)
The main prerequisite for 8701 is good
understanding of undergrad real analysis, such as our 5615H5616H or
equivalent, with substantial experience writing
proofs . Courses named Advanced Calculus are
insufficient preparation. On another hand, there is no assumption of
substantial previous experience with complex analysis, in light of the
peculiarities of undergrad math curricula in the U.S.
Students coming into this course should have a range of
experience in proof writing, not only in a previous course
in analysis, but also in abstract algebra, rigorous linear
algebra, and some pointset topology. All these play significant roles
in 87012, both directly, and in terms of mathematical maturity and vocabulary.
Coherent writing is essential. Contrary to some myths,
the symbols do not speak for themselves.
Prerequisite for 8702: 8701 or equivalent.
Grades fall and spring will be determined
by inclass midterms , scheduled as below. You are
not competing against other students in the course, and I
will not post grade
distributions. Rather, the grade regimes are roughly
90100 = A, 7590 = B, 6575 = C, etc., with finer gradations of
pluses and minuses. So it is possible that everyone gets a "A", or
oppositely. That is, there are concrete goals, determined by what
essentially all mathematicians need to know, and would be happy
to know.
There will be homework/example assignments preparatory to exams,
as scheduled below, on which I'll give feedback about mathematical
content and writing style. The homeworks will not directly contribute
to the course grade, and in principle are optional, but it would
probably be unwise not to do them and get feedback. No late
homeworks will be accepted. Homework should be
typeset, presumably via (La)TeX. If we are meeting in person,
a paper copy is good. Or email a PDF to me. I will post discussions of
the homework/examples prior to the exams. Many of the examples will
resemble prior years', since, after all, there are just a relative few
central examples and ideas. If you find things in prior years' example
discussions, or elsewhere on the internet, or in books, cite
. Also, collaboration with other people is fine, and acknowledge .
This course is not a gauntlet to be run. The course is about
increasing awareness and
exposure to important, useful (also crazy and entertaining)
ideas, so that in the future when they show up (seemingly out of the blue?) in
your work, you can recognize them and act accordingly.
Text will be PDFs posted here, similar to those from previous years.
202122
MWF, 2:303:20, Vincent 207
As in all past years, I strongly encourage people who're ill
not to come to class.
Currently everyone in a classroom is required to
wear a mask (and "wear" means "properly"). Btw, I have been
vaccinated, and take precautions. I strongly encourage
everyone to get vaccinated, if you've not already.
In fact , given the need to accommodate the
inevitable quarantining of people exposed to COVID, and to accommodate
people who have the inevitable symptoms of other bad things... it
seems that there is a strong likelihood of the course turning to
allonline (=Zoom "meeting" in which I screenshare, and talk... and
save PDFs of my writing.) This is far less than ideal... but better
than getting seriously ill or making other people seriously ill or
... An apparent necessity to funnel my presentations through my iPad,
etc., reduce the situation to what I'd done last year purely online
anyway, so there's scant point to be in the friggin' room for it... Sadly.
Office hours MWF after class, in the same room, and email anytime:
I do like talking about math, and answering questions that I already
knowtheanswersto is no burden at all. :)
The notes and homework/examples are roughly copies of the 202021
versions, as placeholders, being updated as we go. Other bits may be
added, as we go, depending on what seems good to include. Prior years'
documents may be updated to correct errors, but will have stable URLs
(to the extent I can control it!)
 00 Intro
[ updated
Sunday, 13Sep2020 15:34:58 CDT]
 01 complex numbers
[ updated
Sunday, 13Sep2020 15:36:07 CDT]
 02 power series
[ updated
Tuesday, 15Sep2020 13:58:43 CDT]
 03 exponential, cosine, sine
[ updated
Sunday, 13Sep2020 15:35:39 CDT]
 Examples 01
[ updated Sunday, 12Sep2021 17:59:28 CDT]
... Discussion 01
[ updated Wednesday, 22Sep2021 18:01:59 CDT]
 04 Cauchy's results and
corollaries: basic complex analysis
[ updated Friday, 16Oct2020 16:33:37 CDT]
 05 the Gamma function [ updated Sunday, 25Oct2020 17:09:26 CDT]
 06
fixedpoint lemma, implicit and inverse function theorems [
updated Sunday, 01Nov2020 17:01:21 CST]
 07 Riemann sphere [ updated Tuesday, 24Nov2020 16:07:24 CST]
 08
conformal mapping [ updated Tuesday, 01Dec2020 13:33:16 CST]

Examples 04 [
updated Wednesday, 25Nov2020 16:17:23 CST]
... Discussion
04 [ updated Saturday, 05Dec2020 15:57:08 CST]
 09
product sine [
updated Sunday, 31Jan2021 17:46:13 CST]

10 partial fractions [ updated Thursday, 19Nov2020 15:12:00 CST]

11 harmonic functions [ updated Wednesday, 03Feb2021 15:35:07 CST]

12 Hadamard products [ updated Wednesday, 03Feb2021 15:35:32 CST]
 13
intro to elliptic functions [ updated Wednesday, 24Feb2021 13:08:24 CST]
 14
intro to elliptic modular forms [
updated Tuesday, 09Mar2021 16:23:17 CST]
 15 intro to Riemann
surfaces [draft] [ updated Thursday, 22Apr2021 18:49:40 CDT] branched/ramified
covers, projectivization/compactification, genus,
RiemannHurwitz formula, hyperelliptic curves, Fermat curves, ...
 Examples 08
[ updated Thursday, 08Apr2021 17:44:53 CDT]
...
Discussion 08
[ updated Wednesday, 21Apr2021 11:30:21 CDT]
 ...
Exam and homeworkexample schedule, fall 2021
Sunday 
Monday 
Tuesday 
Wednesday 
Thursday 
Friday 
Saturday 
  
Sept 08   Sept
10  
 Sept 13   Sept
15   Sept 17 hmwk 01  
 Sept 20   Sept
22   Sept 24 exam 01  
 Sept 27   Sept
29   Oct 01  
 Oct 04   Oct
06   Oct 08  
 Oct 11   Oct 13 
 Oct 15 hmwk 02  
 Oct 18   Oct
20   Oct 22 exam 02  
 Oct 25   Oct
27   Oct 29  
 Nov 01   Nov
03   Nov 05  
 Nov 08   Nov
10   Nov 12 hmwk 03  
 Nov 15   Nov
17   Nov 19 exam 03  
 Nov 22   Nov
24  Thanksgiving 
Nov 26  
 Nov 29   Dec 01   Dec 03 hmwk 04  
 Dec 06   Dec 08
  Dec 10 exam 04  
 Dec 13   Dec 15 last
class    
202021
MWF 2:303:20
Text will be PDFs posted here, similar to those from 201415.
PDFs of handwritten stuff from lectures
 00 Intro
[ updated
Sunday, 13Sep2020 15:34:58 CDT]
 01 complex numbers
[ updated
Sunday, 13Sep2020 15:36:07 CDT]
 02 power series
[ updated
Tuesday, 15Sep2020 13:58:43 CDT]
 03 exponential, cosine, sine
[ updated
Sunday, 13Sep2020 15:35:39 CDT]
 Examples 01
[ updated Wednesday, 09Sep2020 17:24:52 CDT]
... Discussion 01
[ updated Thursday, 17Sep2020 13:43:32 CDT]
 04 Cauchy's results and
corollaries: basic complex analysis
[ updated Friday, 16Oct2020 16:33:37 CDT]
 Examples 02
[ updated Wednesday, 23Sep2020 19:21:55 CDT]
... Discussion 02
[ updated Friday, 16Oct2020 16:30:11 CDT]
 05 the Gamma function [ updated Sunday, 25Oct2020 17:09:26 CDT]
 06
fixedpoint lemma, implicit and inverse function theorems [
updated Sunday, 01Nov2020 17:01:21 CST]
 07 Riemann sphere [ updated Tuesday, 24Nov2020 16:07:24 CST]
 08
conformal mapping [ updated Tuesday, 01Dec2020 13:33:16 CST]

Examples 04 [
updated Wednesday, 25Nov2020 16:17:23 CST]
... Discussion
04 [ updated Saturday, 05Dec2020 15:57:08 CST]
 09
product sine [
updated Sunday, 31Jan2021 17:46:13 CST]

10 partial fractions [ updated Thursday, 19Nov2020 15:12:00 CST]

11 harmonic functions [ updated Wednesday, 03Feb2021 15:35:07 CST]

12 Hadamard products [ updated Wednesday, 03Feb2021 15:35:32 CST]
 13
intro to elliptic functions [ updated Wednesday, 24Feb2021 13:08:24 CST]
 14
intro to elliptic modular forms [
updated Tuesday, 09Mar2021 16:23:17 CST]
 15 intro to Riemann
surfaces [draft] [ updated Thursday, 22Apr2021 18:49:40 CDT] branched/ramified
covers, projectivization/compactification, genus,
RiemannHurwitz formula, hyperelliptic curves, Fermat curves, ...
 Examples 08
[ updated Thursday, 08Apr2021 17:44:53 CDT]
...
Discussion 08
[ updated Wednesday, 21Apr2021 11:30:21 CDT]
 ...
Exam and homeworkexample schedule, spring 2021:
Sunday 
Monday 
Tuesday 
Wednesday 
Thursday 
Friday 
Saturday 
  
Jan 20   Jan 22  
 Jan 25   Jan
27   Jan 29 hmwk
05  
 Feb 01   Feb 03
  Feb 05 exam 05  
 Feb 08   Feb 10
  Feb 12  
 Feb 15   Feb 17
  Feb 19  
 Feb 22   Feb 24 
 Feb 26 hmwk 06  
 Mar 01   Mar 03
  Mar 05 exam 06  
 Mar 08   Mar 10
  Mar 12  
 Mar 15   Mar 17
  Mar 19  
 Mar 22   Mar 24
  Mar 26 hmwk 07  
 Mar 29   Mar 31
  Apr 02 exam 07  
Spring break  Apr 05   Apr 07
 
Apr 09  
 Apr 12   Apr 14
  Apr 16  
 Apr 19   Apr
21   Apr 23 hmwk 08  
 Apr 26   Apr 28
  Apr 30 exam 08  
 May 03 last class  
   
201415
 Outline of course content
[ updated
Sunday, 17Aug2014 17:14:53 CDT]
 Examples 01
[ updated
Thursday, 04Sep2014 11:36:30 CDT]
 00 intro
[ updated
Sunday, 07Sep2014 11:25:22 CDT]
 01 complex numbers
[ updated
Sunday, 07Sep2014 11:08:57 CDT]
 02 power series
[ updated
Tuesday, 14Feb2017 16:03:30 CST]
 03 exp sin cos
[ updated Wednesday, 17Sep2014 15:41:20 CDT]
 04 Cauchy's results and
corollaries
[ updated Wednesday, 17Sep2014 15:44:56 CDT]
 Examples 02
[ updated
Wednesday, 17Sep2014 08:35:43 CDT]
 Midterm 01
[ updated
Wednesday, 24Sep2014 08:01:48 CDT]
 Examples 03
[ updated
Sunday, 28Sep2014 09:36:51 CDT]
 ...
 Examples 04
[ updated
Thursday, 16Oct2014 13:02:06 CDT]
 Review examples 05
[ updated
Wednesday, 05Nov2014 10:58:15 CST]
 Midterm 02
[ updated
Saturday, 01Nov2014 12:18:52 CDT]
 05 More basics: max
modulus, open mapping, Rouché
[ updated
Monday, 03Nov2014 13:00:57 CST]
 Examples 06
[ updated
Friday, 07Nov2014 16:44:25 CST]
 05c Holomorphic
inverse function theorem
[ updated
Thursday, 20Nov2014 14:31:21 CST]
 06 The Riemann
sphere, complex projective line, linear fractional
transformations
[ updated
Saturday, 22Nov2014 13:11:06 CST]
 07
Elementary conformal mappings
[ updated
Sunday, 23Nov2014 13:46:25 CST]
 Examples 07
[ updated
Friday, 21Nov2014 13:05:21 CST]
 Midterm 03
[ updated
Monday, 01Dec2014 09:28:59 CST]
 Sample final exam
[ updated
Thursday, 11Dec2014 08:37:58 CST]
 Actual final exam
[ updated
Tuesday, 09Dec2014 10:48:42 CST]
 08 Partial fraction expansions, product expansions:
 Examples 08
[ updated
Sunday, 01Feb2015 17:57:47 CST]

09 Prime number theorem and Riemann's zeta function
[ updated
]
 Example discussion 08
[ updated
Sunday, 01Feb2015 17:57:24 CST]
 ...
 Examples 09
[ updated
Thursday, 12Feb2015 10:48:41 CST]
 Midterm 04
[ updated
Monday, 02Mar2015 09:20:40 CST]
 Elliptic integrals, elliptic functions, elliptic modular forms
 Examples 10
[ updated
Tuesday, 24Mar2015 10:55:27 CDT]
 Midterm 05
[ updated
Thursday, 02Apr2015 08:06:22 CDT]
 Supporting analytical notes
 Intro to Riemann surfaces
[ updated
Monday, 20Apr2015 13:13:13 CDT]
 Examples 11
[ updated
Tuesday, 14Apr2015 10:31:51 CDT]
 Midterm 06
[ updated
Wednesday, 29Apr2015 12:05:10 CDT]
 ...
Older notes
Unless explicitly noted otherwise, everything here, work
by Paul Garrett, is licensed
under a Creative
Commons Attribution 3.0
Unported License.
...
[ garrett@math.umn.edu ]
The University of Minnesota explicitly requires that I
state that "The views and opinions expressed in this page are
strictly those of the page author. The contents of this page have not
been reviewed or approved by the University of Minnesota."