Complex Analysis, 4th Edition, by Serge Lang, published by
Springer-Verlag, 1999.
Course Outline:
Hopefully we'll cover material in the textbook up through Chapter 6.
Grading:
Homework ------ 30%
Midterm Exam --- 30%
Final Exam ----- 40%
The final exam is scheduled for 8-9:50 am, Thurs, May 8,
2003. Students are strongly encouraged to work together on
homework assignments. Late homework will be accepted only
with the advanced permission of the instructor.
Prove, via a simple counter example, that C[a,b] is not complete
with respect to the L-2 norm. (due 2/7/03)
Prove that C[a,b] is complete with respect to the L-infinity, or sup,
norm. Also, prove that C(R) is not complete with respect to the
L-infinity norm. (recommended for math students)
Homework for 2/5/03:
Provide a rigorous proof of the ratio test (recommended for math
students), making use of comparison with a geometric series.
Comment on Problem 6 of the Midterm Takehome Exam: The key result that
you need is Theorem 7.1 (Cauchy formula on a disk) on p. 126 of the text.
This result holds for a disk centered at z0 (which you can take to be 0)
for ***any*** radius r, since f is assumed to be holomorphic on all of the
complex plane. To get the needed formulas for the derivatives, you can either
(i) repeatedly differentiate the Cauchy formula; or (ii) expand 1/(w-z)
as a geometric series as on p. 128, interchange integration and summation
to get the expression at the bottom of p. 128, and relate the coefficients
of the power series to derivatives of f evaluated at z0. Once you have
a Cauchy-like derivative formula, proceed as in Monday's class notes.
Homework for Friday, 3/21/03: Work out the details of both
approaches (i) and (ii). I know that some of you already did this for
approach (ii) in Problem 3. If this is the case, please recopy the
relevant material and hand it in together with part (i).
Section V.3, p. 170: Exercises 1ab,2,5,6,9. Recommended: #10.
Homework for 4/21/03:
Section VI.1, p. 186: Exercises 18,19
Section VI.2, p. 204: Exercises 1a,5ab,10,11,14a,18
Use Laplace transforms & the Bromwich inversion formula to solve
the ODE IVP u'(t) - u(t) = sin(t), t>0; u(0)=2.
Carefully explain what happens, in the limit, to all the contour
integral terms. You will first need to show that the Laplace transform
of sin(t) is 1/(s^2 + 1).