# Complex Variables I

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MATH-GA.2450-001

Fall 2014, Thursdays 5:10-7:00pm, WWH Room 102

Instructor: Olof
B. Widlund

** Coordinates **

Office: WWH 612

Telephone: 212-998-3110

Office Hours: Tuesdays 5:00-5:45pm and Wednesdays 3:00-4:00pm.

e-mail: widlund at cims.nyu.edu

Contrary to what is said in the ''Course Descriptions Academic Year 2014-15'',
students can take this course the same semester as Introducion to Mathematical
Analysis I.

** Requirements **

Regular homework assignments. It is important that you do the homework yourself, but when you get stuck,
I encourage you to consult with other students
or me, to get help when necessary. However, when you get help,
it's important to acknowledge it in writing. Passing off other people's
work as your own is called plagiarism and is not acceptable.
Please staple everything together and order the problems
in the same order as given.

** Final Exam **

There will be an in-class final exam in WWH 102 on Thursday December 18
at 5:00pm. You will not be able to bring any books or electronic devices
to the final but you can bring two regular-sized sheets of paper covered
on both sides with notes written by yourself. At the final, you will have to
sign an attendance sheet and you must also bring
your NYU ID.
Here are seven questions that might help in
preparing for the final.

** Homework Assignments **

Homework scores will be made available through the NYU Classes system.
Home work assignments will not be available through that system.
You can download the assignments in pdf format from this home
page.

Homework 1 Assigned September 7,
due Friday September 19, at 12 noon.

Homework 2 Assigned September 17,
due Friday September 26, at 12 noon.

Homework 3 Assigned September 22,
due Friday October 3, at 12 noon.

Homework 4 Assigned September 26,
due Friday October 10, at 12 noon.

Homework 5 Assigned October 7,
due Friday October 17, at 12 noon.

Homework 5 Assigned October 7,
due Friday October 17, at 12 noon.

Homework 6 Assigned October 15,
due Friday October 24, at 12 noon.

Homework 7 Assigned October 22,
due Friday October 31, at 12 noon.

Homework 8 Assigned October 30,
due Friday November 7, at 12 noon.

Homework 9 Assigned November 6,
due Friday November 14, at 12 noon.

Homework 10 Assigned November 13,
due Friday November 21, at 12 noon.

Homework 11 Assigned December 4,
due Friday December 12, at 12 noon.

** Lectures **
September 4: Historical remark about 16th century Italian mathematics;
solving x^3=3px+2q to find a real root requires us, at times,
to use arithmetic with
complex numbers. Some contribution of Euler in the 18th century including
the use of i (i^2=-1) and polar form (polar coordinates) to represent complex
numbers. Extending the set of real numbers to the set of complex numbers. How
to add and multiply complex numbers by representing them as pairs of reals
and special formulas. An
alternative approach using special 2-by-2 matrices and matrix adition and
multiplication. Dividing by complex numbers and complex conjugation. Basic
inequalities including the triangle inequality. Polar form, the argument and
modulus of a complex number. How to multiply two complex numbers and how
to divide complex numbers using the polar form. Vector representation of
complex number. Two excercises; the second concerns Cauchy's inquality and
will be revisited on 9/11.

September 11: More details of the proof of Cauchy's inequality. Solving two
elementary geometric problems using complex numbers. The stereographic
projection and how to understand the extended complex plane, in particular
infinity. Open sets, their boundaries, closed sets, domains, and regions.
Limits of functions and continuity. Derivatives of complex-valued functions
of a complex variable. Cauchy-Riemann's equations which must be satisfied in
order for f(z) to have a derivative and which provide criteria to decide if
a derivative exists. Harmonic funtions; the real and imaginary parts of
f(z) are harmonic if the derivative exists. Computing derivatives after
introducing polar coordinates and the Cauchy-Riemann's equations in polar
coordinates.

September 18: More on the stereographic projection; a proof that those
three point indeed lie on a straight line. The distance between two points on
the sphere in terms of the corresponding points in the complex planes. Limits
of sequences and how to handle limits of functions which equal infinity and
when the argument of the function goes to infinity, or both. Analytic
(holomorphic) functions and how they are uniquely determined by their values
on certain subsets of their domain of definition. Zeros and poles of rational
functions including zeros and poles at infinity. A few words on analytic
continuation. The exponential function and sine and cosine of a complex
variable.

September 25: Hadamard's formula for the radius of convergence of
power series. The radius of convergence of the power series expansion
of the exponential function. Limits of sequences and partial sums and
the definition of lim sup of sequences. Absolute convergence of sequences.
Convergence and uniform convergence of sequences of functions; a uniformly
convergent sequence of continuous functions have a continuous limit. A proof
of Hadamard's result and proof of uniform convergence of power series. A proof
of the analyticity of any function defined by a convergent power series
by using the definition of the derivative and Hadamard's formula.
Periodicity of the complex exponential function. Its inverse which is a
multi-valued function. Branches of the logarithmic function, branch points and
branch points. Analyticity of branches of the logarithmic function.

October 2: Introduction to integrals, closely following sections 41-47
of the ninth edition of the Brown-Churchill text.

October 9: Existence of antiderivatives, integrals around closed contours
that vanish, and the independence of iintegrals on the path between the end
points of paths.
The Cauchy-Goursat theorem for rectangles; we do not need continuity
of the derivative of f(z). Comments on extensions to more general domains such
as star domains and circles. Multiply connected domains, contours that
crosses themselves, and deformation of paths. Caucy's integral formula and
extentions which provide formulas
for derivatives. Simple examples of how to evaluating some integrals by using
these formulas.

October 16: Extensions of Cauchy's integral formula to represent
derivatives of analytic functions. Legendre polynomials; they are orthogonal
polynomials. A proof of the fact that an analytic function has derivatives
of all order. A proof that the real and imaginary parts of any analytic
function are infinitely differentiable. Liouville's theorem and the fundamental
theorem of algebra. The maximum modulus principle. Taylor series expansions
of analytic functions. A few words on Laurent expansions.

October 23: Taylor series expansions of analytic functions and a proof
of the convergence os these series. The coefficients can be found by computing
higher and higher order derivatives but often by more elementary means such
as as products of series expansions of functions for which the expansions are
well known. Laurent expansions and how to find their coefficients using
formulas obtained by using Cauchy's integral formula and its extensions
of by manipulating Laurent series just as for Taylor expansions. The importance
of the term with a first negative power of z-z_0; the other terms have
well-known anti-derivatives. A proof of the correctness of the formulas
for the Laurent expansion.

October 30: Integrating as in Sections 85-90 of the Brown-Churchill text
book illustrated by solving problems that fit into the different problem
classes.

November 6: More about computing definite integrals, in particular,
the familiy discussed in Section 92 of Brown-Churchill. How to compute
infinite sums using residues; not covered in the text. Inverse Laplace
transforms. Local properties of analytic functions, removable singularities,
poles, and isolated essential singularities. The Casorati-Weiserstrass theorem.
The uniqueness of analytic functions: if two analytic functions defined in the
same open set D coincide in a subset, they are the same in all of D.

November 13: Riemann's theorem. The argument principle and winding numbers.
Rouche's theorem. Linear fractional transformations. Their fixed points.

November 20: A discussion of the Gamma function and how it can be displayed
with a handout of two plots from a classical book on special functions.
Conformal maps and how they preserve angles and map small squares into small
squares with slightly curved sides. Some mappings that do not belong to
the family of linear fractional transformations: w = z^2, the exponential
function, w= sin(z), sinh(z), squareroot(z), and z^{1/n}. Riemann surfaces.
(Most of the lecture followed the Brown-Churchill book quite closely.)

November 27: Thanksgiving Day.

December 4: Formulation of Riemann's mapping theorem. Conformal maps have
inverse maps that also are conformal. The Joukowski transformation and its
critical points. The mapping of a circle of radius square-root(5/2) and
centered at (i-1)/2 under this transfomation. A flawed computation, to be
corrected, which meant to have shown that the curve mapping this circle
passes the origin. The picture is one of a fat airfoil. The Joukowski map
maps the upper half plane onto the extended complex plane except for the
interval [-1,1] on the real axis; this is a one-to-one map. The lower half
place is also mapped onto the same set. This is a reflection of the fact that
for most every point in the w-plane there are two pre-images in the z-plane.
Dirichlet and zero Nuemann boundary conditions for Laplace's equation.
Solving the Dirichlet problem in a circular disk by Poisson's formula; this
discussion to be continued. The idea is then to combine this formula with
a conformal map to obtain a solution of Laplace's equation on other domains.
Solving Laplace's equation on a half space; this provides an alternative
to using Poisson's integral formula. Note that except for the Joukowski
map and the Riemann mapping theorem, this material can be found in the text
book.

December 11: Constructing a harmonic function v(x,y) such that
u(x,y) + iv(x,y) is an analytic function and u(x,y) is a given harmonic function. Under an analytic mapping any harmonic function is mapped onto another
harmonic function. A derivation of Poisson's integral formula and properties
of the kernel. A discussion of six of the sample final problems.

December 18: In class final.

** Text Book **

Complex Variables and Applications. by J.W. Brown and R.V. Churchill, Ninth
Edition, McGraw Hill.