Description

Welcome to MTH 829 Complex Analysis. Please follow the link below to find the syllabus. Announcements about the course will be posted here. The on-line forum for asking and answering questions can be found at the Q&A tab above.

To use the forum you will need to register at the link you will have received in your msu email. If you have not used piazza before with your msu email, you will be asked to create a password the first time you log in.

General Information


Announcements

SIRS and Course Meeting 4/28
4/25/17 3:41 PM

I will be in the classroom at our usual meeting time  on Friday 4/28 to return exams and homework.  In the meantime, please fill out the SIRS form for this course:

 https://sirsonline.msu.edu.

In particular, I very much appreciate constructive comments on how the course could be improved as well as specific comments regarding what you think is working well.

Links to Old Qualifiers
4/17/17 3:53 PM

Two old qualifying exams may be downloaded from the resources section of this Piazza page

Homework 9 -- due April 14
4/07/17 12:31 PM

From Sarason Chapter X: 12.2, 12.3, 12.7, 12.8, 16.1, 16.5, 19.3, 19.4 

Homework 8 -- due April 5
3/28/17 4:07 PM

From Sarason Chapter X: 10.3, 10.4, 10.6

Additional problems:

Problem 1: Let b>1 and 1<a<b1.  Find a formula for

0xa1+xb

and prove that it is correct.

Problem 2: Compute 

1x4+1eikxdx

for kR.  Prove that your answer is correct.

Homework 7 -- due March 24
3/17/17 3:53 PM

From Sarason:   IX.5.3, IX.17.1 (Hint: fix z_0 and consider the set of all points that can be reached from z_0 along a polygonal path.  Prove that this set is open and closed.)

Also work the following three problems: 

Problem 1: Let G={x+iy:1<x<1}. Suppose that f is holomorphic on G{0} and that limzf(z) exists.  Prove that 

limRRR(f(x+iy)f(x+iy))dy = 2πres0f

for any x(0,1).

Problem 2: Let G=C(,0] denote the slit plane.  Let  f be a holomorphic function on G. Suppose that

  1. limzf(z)=0, and
  2. limz0|z|f(z)=0.

Suppose further that for each x(,0) the two limits

limη0f(x+iη) = ϕ+(x)

and

limη0f(x+iη) = ϕ(x)

exist, with the convergence being locally uniform in x(,0). Prove that

f(z) = limR12πiR01xz(ϕ+(x)ϕ(x))dx

for any zG. 

Problem 3: Use the previous problem to derive the formula

1z = 1π01x+z1xdx,

where  denotes the principle branch of the square root.

Correction and a link
3/14/17 10:51 AM

In yesterday's lecture I state the last theorem incorrectly.  The correct statement is as follows:

Thm. Let γ:[a,b]C be a piecewise C1 closed curve such that for some t0[a,b] we have γ(t)γ(t0) for all tt0.  Then the interior of γ is non-empty.

The restriction that γ(t0)γ(t) for all tt0 was used implicitly in the proof, since we assumed that the segment looked at was traversed only once.  It is required to rule out a curve such as a line segment traversed forwards and backwards, which has no interior.

If you are curious, a proof of the Jordan Curve theorem for piecewise smooth curves can be found here:

https://www.jstor.org/stable/2316660

The general theorem, for continuous curves,  is more difficult to prove.

Exam 1 revisions -- due March 3
3/02/17 1:09 PM

Please turn in revised solutions to problems 2 and 6 on the exam  along with your original exam in class tomorrow, March 3. 

Homework 6 -- due March 13
2/26/17 9:18 PM

From Sarason Ch. VIII: 1.1, 2.1, 4.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.2, 12.1, 12.2, 12.3

Staff Office Hours
NameOffice Hours
Jeffrey Schenker
When?
Where?
Rodrigo Bezerra de Matos
When?
Where?

Homework

Homework
Due Date
Feb 24, 2017
Feb 15, 2017
Feb 6, 2017
Jan 30, 2017
Jan 20, 2017

Homework Solutions

Homework Solutions

Lecture Notes

General Resources