Math 715: Applied Complex Analysis
Days & Times Room Instructor Meeting Dates
------------------------------------------------------------------------
TBA DMS238 Eric Olson 08/27/2012 - 12/19/2012
Course Information
- Instructor:
- Eric Olson
- email:
- ejolson at unr edu
- Office:
- Monday, Wednesday and Thursday 1pm DMS 238 and by appointment.
- Homepage:
- http://fractal.math.unr.edu/~ejolson/701
- Texts:
-
Carrier, Krook and Pearson, Functions of a Complex Variable: Theory and Technique, SIAM, 2005.
-
Michael Renardy, Robert C. Rogers, An Introduction to Partial Differential Equations, Texts in Applied Mathematics 13, Springer-Verlag, 1993.
- Other Complex Analysis Books:
- A.I. Markushevic, Theorey of Functions of a
Complex Variable, Second Edition, AMS, 2005.
- Lars Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill, 1979.
- John Conway, Functions of One Complex Variable,
Second Edition, Springer, 1978.
- M.A. Evgrafov, Analytic Functions, Dover, 1966.
Announcements
[26-Aug-2012] Meeting Times
Meeting times are 3pm on Tuesdays in my office.
[21-Aug-2012] Meeting Times
We will meet in my office. I am available after 2pm MTWF. Please send
me an email detailing your teaching schedule and when you are available
so I can find a common time for us to meet.
Homework
Homework 1 for September 4
0. Finish analyzing the radius of convergence of the power series
where 4a_2=-a_0, 12a_3=-a_1 and 4(k+2)a_k+2=(k-2)a_k.
1. Read chapter 4 from Boyce and DiPrima on power series solutions
to linear second order ordinary differential equations.
2. Review the proof of the Cauchy integral formula from page 37 of
Carrier Krook and Pearson.
3. Review the proof of the Maximum Modulus Theoarm from page 42 of
Carrier Krook and Pearson.
4. Read about analytic continuation on page 64 of Carrier, Krook
and Pearson.
5. Explain how easy or difficult it is to understand pages 55 - 65
in Carrier, Krook and Pearson.
Homework 2 for September 11
1. Boyce and Diprima page 166 problem 16.
2-3. Read section 2.1.1 and 2.2.1 in Rogers and Renardy.
4. Carrier, Krook and Pearson page 56 exercise 1.
5. Boyce and DiPrima page 190 problem 2.
6-8. Try to find a power series solution of (4-x^2)y''+2y=0 expanded
around the points x_0=1, x_0=2 and x_0=3.
9. Use problem8 on page 37 of Carrier, Krook and Pearson to prove
the power series we obtained is really a solution to the ODE.
Homework 3 for September 18
1. Use the technique of solving for r such that |a_n| <= Cr^n
to work problem 8 from last week. In particular, find a lower
bound on the power series solution of
(4-x^2) y'' + 2 y = 0
expanded about x0 = 3.
2. Finish Carrier, Krook and Pearson page 56 problem 1 for the
annulus pi < |z| < 2 pi.
3. The point x0 = 2 is a regular singular point of
(4-x^4) y'' + 2 y = 0
Find a solution to this equation. Determine whether there are
two linearly independent solutions at x0 = 2.
4. Continue working on page 190 problem 2 of Boyce and DiPrima.
Check that a0<>0, a2=-a0/4!, a4=a0/6! give the first three
non-zero terms the power series solution for r = 1.
5. First the first three terms of the power series solution to
page 190 problem 2 of Boyce and DiPrima for r = -1.
6-7. Read sections 2.2.2 and 2.2.3 in Rogers and Renardy.
Homework 4 for September 25
1. Let h:C->C be such that h'(z)->L as z->a. Prove or disprove
the following claims:
a. There exists M such that h(z)->M as z->a.
b. Let H(z)=M for z=a and h(z) otherwise. Then H'(z) exists.
c. H'(z)=L.
Let h:R->R be such that h'(x)->L as x->a. Do any hypothesis
need to be added so the claims in the previous question hold?
2. There is a theorem which states that if f has an isolated
singularity at a in C then either
a. the discontinuity is removable, or
b. __________________________________.
What goes in the box?
3. Suppose x0 is a singular point of the ordinary differential
equation p(x)y''+q(x)y'+r(x)y=0. For s not equal x0 define
x->y(x,s) to be the solution to this differential equation
satisfying the initial conditions y(s,s)=a0 and y'(s,s)=a1.
a. Is it possible to define a function f in a neighborhood
of x0 by formula
f(x) = lim y(x,s)
s->x
and, if not, is there an additional hypothesis that can
be added so f is well defined?
b. Does y=f(x) satisfy the differential equation?
c. Does f(x)=a0 and f'(x)=a1?
4. Continue reading Renardy and Rogers 2.2.2 and 2.2.3.
Homework 5 for October 2
1. What happens with the Laurent series if the curve C passes
through the point z?
2. Consider question 1 when the curve is smooth.
3. Consider question 1 when z is a corner point of C.
4. What is a corner point?
5. Read about principle value integrals in Carrier, Krook and
Pearson page 39 and pages 413-418.
Homework 6 for October 11
1. Print my solution to to page 36 # 8 in Carrier, Krook and
Pearson. Read it and check for errors.
2. Write out a proof of item 1 in Rogers and Renardy page 47.
3. Verify the that C_alpha=(1/alpha!) D^alpha f(x) in item 3
of Rogers and Renardy page 47.
Homework 7 for October 18
1. Suppose K is compact, fn(x) are continuous and there is M
so large that
Sum {|fn(x)| n=1..infinity} <= M for all x in K
Prove or disprove the claim that
Sum {|fn(x)| n=1..N}
converges uniformly for x in K as N -> infinity.
2. Suppose K is compact, Pn(x) are polynomials of degree n and
there is M so large that
Sum {|Pn(x)| n=1..infinity} <= M for all x in K
Prove or disprove the claim that
Sum {|Pn(x)| n=1..N}
converges uniformly for x in K as N -> infinity.
3. Let
f(x) = exp(-1/x^2) if x<>0
= 0 if x=0
Prove or disprove that
f(x)>0 for x<>0
d^nf(x)/dx^n=0 for x=0 and n=1,2,3....
4. Find another example of a function g(x) such that
g(x)>0 for x<>0
d^ng(x)/dx^n=0 for x=0 and n=1,2,3....
5. Can Theorem 2.17 in Rogers and Renardy be rewritten as
w
C (Omega) = Intersection Union Union Intersection Cm,r(y)
compact S in Omega m>0 r>0 y in S
6. Read proof of 2.15 and 2.17 in Rogers and Renardy.
7. Think about conformal maps.
8. Look up the Ascoli theorem.
Homework 8 for October 25
1. Work problem 3 of page 123 and problems 1 and 3 on page 134
of Carrier, Krook and Pearson.
2. Answer the following questions about the pictures:
(i) Why do the black lines meet at perpendicular?
(ii) Why does the green dot move backwards?
Homework 9 for November 8
1. Page 44 problems 4, 5, 6 of Carrier, Krook and Pearson.
2. Read Theorem 2.17 on page 48 in Renardy and Rogers and
rewrite in terms of a single complex variable.
3. Read pages 49 to 60 in Renardy and Rogers.
Homwork 10 for November 15
1. Prove that a closed and bounded set in C^n is compact. That
is, show that if W is closed and bounded, then any open cover
admits a finite subcover.
2. Fill in details of the proof of Theorem 2.17 for the forward
implication in Renardy and Rogers.
Final Exam
The final exam is TBA.
Equal Opportunity Statement
The Mathematics Department is committed to equal opportunity in education
for all students, including those with documented physical disabilities
or documented learning disabilities. University policy states that it is
the responsibility of students with documented disabilities to contact
instructors during the first week of each semester to discuss appropriate
accommodations to ensure equity in grading, classroom experiences and
outside assignments.
Academic Conduct
Bring your student identification to all exams. Work independently on
all exams and quizzes. Behaviors inappropriate to test taking may disturb
other students and will be considered cheating. Don't talk or pass notes
with other students during an exam. Don't read notes or books while taking
exams given in the classroom.
You may work on the programming assignments in groups of two if desired.
Homework may be discussed freely. If you
are unclear as to what constitutes cheating, please consult with me.
Last updated:
Tue Aug 21 12:11:04 PDT 2012