MAT 342, Applied Complex Analysis
Fall 2016
Christopher Bishop
Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631
bishop at math.stonybrook.edu
Time and place: MWF 10:00-10:53am, MATH P-131
(ROOM CHANGED from Physics P-112)
academic calendar
Exam calendar
Grader: Jack Burkart, jack.burkart at stonybrook.edu, office hours
Wed 11am-12pm in 5-125b (Math Tower); TuTh 1pm-2pm in Math Learning Center
(S-level = basement in Math Tower)
FINAL EXAM : Tuesday, December 20, 2:15-5:00pm in usual room (P-131 Math Building).
REVIEW SESSION : Friday, December 16, 10-11am in usual room (P-131 Math Building).
practice first midterm
This is 50 questions. The actual midterm will have exactly the same
format, although the questions will be changed. Most of the questions
are not hard, but you will have to move quickly to finish the whole
exam. We will go over the practice in class on Monday, Oct 10.
The midterm is intended to cover Chapters 1-4. A few questions on
the midterm refer to things that we have not yet covered in class,
but will cover before the midterm (if we don't, then I will make sure
this material in not on the actual midterm).
answers for practice first midterm
answers for first midterm
histogram of results for first midterm
practice second midterm The actual
midterm is Wednesday, October 16, and will look simliar to the
practice, but may contain a few questions with a different format.
The practice exam was written quickly and may have some typos. I will
try to post an answer key by Sunday (Oct 13) evening.
answers for practice second midterm
answers for second midterm
histogram of results for second midterm
Final Exam: Tuesday, December 20, 2016 2:15pm-5:00pm, location to be announced (probably during last week of classes)
The textbook "Complex Variables and Applications" by J.W. Brown
and R.V. Churchill, 9th edition. The maerial covered in earlier
editions should be similar, but sections and exericses might be
numbered differently. All assignments will be based on 9th edition,
so if you are using a different edition, it is your responsibility
to make sure you do the correct work.
8th edition
This is an introduction to functions of a complex variable and
emphasizes developing computational skill with complex numbers
(complex arithmetic, power series manipulation, evaluation of
real and complex integrals using residues,...).
It is also a mathematically rigorous course, and most statements
will come with complete proofs. Students will be expected to be
able to do simple proofs and derivations, as well as perform the
calculational skills mentioned above. Topics covered will include
properties of complex numbers, analytic functions with examples,
contour integrals, the Cauchy integral formula, the fundamental
theorem of algebra, power series and Laurant series,
residues and poles with applications, conformal mappings
with applications and other topics if time permits.
An alternative (or sequel) to this course is MAT 536 (previously
numbered MAT 542). This is a first year graduate course in one
complex variable that is offered every Spring. It covers about
twice as much material at MAT 342
and is much more theoretical (all proofs, all the time).
Grades will be based on weekly problem sets (50 points total),
two midterms (50 points each) and a final exam (50 points).
Prof. Bishop's Office Hours: MW 9-10, M 2-3. You should aso
feel free to make appointments for other times, or just stop
by my office.
tentative lecture schedule and
problem assignments
The schedule may change as we proceed,
and this may change the assigments. I will post updated copies here
if that happens.
I updated the homework on Sept 23,to correct a type in the assignment for Sept 26. Problems that were marked as in Section 37 did not exist and were meant to be in Section 36.
Send me email at:
bishop at math.sunysb.edu
(I usually respond faster to this than to my @stonybrook.edu
account)
Send the grader an email at:
jack.burkart at stonybrook.edu
What we did in class:
Mon. Aug 29: introduction, the complex plane,
complex addition and mulitplication, some advantages
of using complex numbers, the field axioms
Wed. Aug 31: review addition and multiplication, real
and imginary parts, complex conjugates, absolute value,
division, polar coordiates, unit complex numbers, powers
of i, triangle inequality, squaring , square roots
Fri. Sept 2: exponential form (with motivation based on
power series), Euler's formula, find powers and roots,
de Moivre's formula, the quadratic formula. Introduction
to planar topology, balls, open and closed sets, boundary
of a set, interior and exterior points, connected set,
bounded versus unbounded, accumulation point.
Wed. Sept 7:
Review planar topology,
functions, domains, real-valued, polynomials, rational
functions, image inverse image, translation, rotation,
dilation, reflection, geometryof z^2, definition of
limit, uniqueness, real and imaginary parts, addition
of limits, multiplication of limits, limits involving
infinity.
Fri. Sept 9:
two definitions of continuity, composition of
continuous functions, real and imaginary parts of
continuous functions, boundedness, definition of
derivative, examples, rules for differentiation.
Mon. Sept 12:
Cauchy-Riemann equations, differentiable implies
CR euations hold, examples, CR equations imply
differentiable.
Wed. Sept 14:
Sections 24,25,26.
Polar form of Cauchy-Riemann equations, definition and
examples of analytic functions, zero deriviative implies
constant function.
Fri. Sept 16 :
Sections 27, 28,29.
Harmonic functions, uniquely determined analytic functions,
the reflection principle.
Mon. Sept 19:
Sections 30, 31, 32, 33. Exponentials and logarithms.
Wed. Sept 21:
Sections 34, 35, 36, 37.
More about logs. Power functions, trigonometric functions.
Fri. Sept 23 :
Sections 38, 39, 40.
Inverse trig functions, hyperbolic functions
Mon. Sept 26:
Sections 41, 42, 43.
Derivatives, integrals, contours.
Wed. Sept 28:
Sections 44, 45, 46, 47.
Contour integrals, examples, branch cuts, upper bounds.
Fri. Sept 30 :
Sections 48, 49
Independence of integral from contour; statement and proof.
Mon. Oct 3:
Sections 50, 51.
Cauchy-Goursat theorem; statement and proof.
Wed. Oct. 5:
Sections 52, 53.
Simply connected and multiply connected domains.
Fri. Oct 7:
Liouville's theorem, fundamental theorem of algebra,
Mon. Oct 10:
Review practice Midterm
Wed. Oct. 12:
Maximum principle, review for Midterm
Fri. Oct 14:
Midterm
Mon. Oct 17:
Introduction to power series, Taylor series for analytic functions
Wed. Oct. 19:
Laurent series
Fri. Oct 21:
Uniform convergence, continuity of limit, limits of integrals
Mon. Oct 24:
Uniform convergence of power series, uniqueness, term-by-term
integration and differentiation, multiplication and division of
series.
Wed. Oct 26:
Isolated singularities, residues, Cauchy's residue theorem
Fri. Oct 28:
Residue at infinity, three types of point singularities, examples
Mon. Oct 31, go over first midterm :
Wed. Nov 2:
Residues at poles, examples
Fri. Nov 4:
zeros of analytic functions, zeros and poles,
Mon, Nov 7 :
removable singularities, essential singularities and poles
Wed. Nov 9
Evaluation of integrals using residues
Fri. Nov 11:
Evaluation of integrals using residues
Mon, Nov 14 :
review for Midterm 2
Wed. Nov 16
Midterm 2
Fri. Nov 18:
Argument Principle, Rouche's theorem
What I plan to do next
Mon. Nov 21:
Class canceled. Problems due today will be due Nov 28 instead.
Mon. Oct 28: Linear and linear fractional transformations,
applications to topology of surfaces and manifolds
Wed Nov 30:
go over second midterm
Fri. Dec 2.
Guest Lecture by Prof Mulhaupt of AMS: complex analysis and quantitative
finance.
Mon. Dec 5:
Guest Lecture by Prof Sutherland: root finding and fractals
Wed. Dec 7:
Conformal mapping
Fri. Dec 9
Review for final.
Tue. Dec 20
final exam, 2:15-5:00pm
Additional links
Link to
history of mathematics