Math 501 Fall 2003

Instructor: John Roe


This is the first part of the Analysis sequence for newly arriving graduate students in Mathematics.
We will study complex-valued functions of a complex variable in the first part of the semester, and Hilbert space
in the second part.

Prerequisites:  A knowledge of calculus, analysis and linear algebra at the undergraduate level, such as is provided by the Graduate Study option in the Penn State mathematics major. We will also take for granted some key properties of integration.  We'll review these at the beginning of the course.  A full-dress treatment of integration, at the graduate level, is contained in Math 502 which you will take next semester.

Meeting Times:  The class meets three times a week, on Mondays, Wednesdays, and Fridays from 11.15 to 12.05 in  140 Fenske ( not 213 Buckhout nor 307 Boucke as were previously scheduled).

Office hours will take place Wednesday 2.15 -- 3.00  and Friday 10.15 -- 11.00.  Students are strongly encouraged to make use of available office hours to discuss any questions or problems that they may have about the course or about mathematics more
generally .

Academic Integrity Statement All Penn State policies regarding ethics and honorable behavior apply to this course. Academic integrity is the pursuit of scholarly activity free from fraud and deception and is an educational objective of this institution.  Academic dishonesty includes, but is not limited to, cheating, plagiarizing, fabricating of information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. For any material or ideas obtained from other sources, such as the text or things you see on the web, in the library, etc., a source reference must be given. Direct quotes from any source must be identified as such. All exam answers must be your own, and you must not provide any assistance to other students during exams. Any instances of academic dishonesty will be pursued under the University and Eberly College of Science regulations concerning academic integrity.

Grading  Your grades for this course will be computed on the basis of weekly homework assignments and a final exam.  Homework assignments will be posted on this web site and handed out in class.  They will be due on Fridays (starting Friday September 12th) and I will aim to return graded homework on the following Mondays.  Each assignment will contain three questions.  Part of my aim in this course is to help you write extended mathematical argument, and therefore these questions will require extended answers (up to two or three pages in length), and you will be graded on the quality and coherence of your exposition as well as on whether you have the "right answer".

Grades will be calculated as follows:
Here are the posted homework assignments Here is a grade sheet for the course.
 

Course outlines are given below.  These are subject to change as the semester progresses.


Complex Analysis, 28 Lectures

  • Textbook: H.A.Priestley, Introduction to complex analysis (OUP), second edition.  This book has now arrived in the bookstore

  •  

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

    Supplemental: J. Stalker, Complex analysis: fundamentals of the classical theory of functions (Springer)
    T. Needham, Visual Complex Analysis, (Clarendon)

    Official Course Description: Various forms of Cauchy's theorem. Cauchy integral formula. Power series, Laurent expansion. Residue calculus and applications. Properties of harmonic functions. Conformal mapping. Riemann mapping theorem (proof as time allows).

    Unofficial Course Description: We will study the behavior of differentiable complex-valued functions f(z) of a complex variable z.  The key idea in the course is that complex differentiability is a much more restrictive condition than real differentiability.  In fact, complex-differentiable functions are so rigid that the entire behavior of such a function is completely determined if you know its values even on a tiny open set.  One understands these rigidity properties by making use of contour integration - integration along a path in the complex plane.

    The theory gains its force because there are plenty of interesting functiuons to which it applies.  All the usual functions - polynomials, rational functions, exponential, trigonometric functions, and so on - are differentiable in the complex sense.  Very often, complex analysis provides the solution to "real variable" problems involving these functions; as someone said, "The shortest path between two real points often passes through the complex domain." Moreover, complex analysis is a key tool for understanding other "higher transcendental functions" such as the Gamma function, the Zeta function, and the elliptic functions, which are important in number theory and many other parts of mathematics.  A secondary aim of this course is to introduce you to some of these functions.

    One of the surprises of complex analysis is the role that topology plays.   Simple questions like "do I choose the positive or negative sign with the square root" turn out to have surprisingly subtle answers, rooted in the notion of the fundamental group of a topological space (which you will be looking at in the Topology and Geometry course parallel to this).  These topological notions eventually culminate in the notion of a Riemann surface as the correct global context for complex analysis.  We will not develop this idea fully, but we will discuss `multiple-valued functions' and their branch points; again, we will try to illustrate how these exotic-sounding concepts help in doing practical calculations.
     
     

    Schedule of Lectures (Provisional)

    1 Introduction: `complex is simpler'.
    Review of power series.
    Notes
    2 Conformal linear transformations.  Holomorphic functions, conformality, Cauchy-Riemann equations. Notes
    3 Examples of holomorphic functions. Notes
    4 Conformal mapping; properties of Mobius transformations; exponentials and powers.
    5 More examples; discussion of integration along paths Notes
    6 Estimates for integrals; Cauchy's theorem for a triangle Notes
    7 Cauchy's theorem for a convex set Notes
    8 More general Cauchy theorem (homotopy form) Notes
    9 The Cauchy package: integral formula, Taylor's theorem, Morera's theorem. Notes
    10 Isolated zeroes, maximum modulus principle, fundamental theorem of algebra Notes
    11 Singularities and the Residue theorem; Notes
    12 Zero counting and applications Notes
    13 Laurent's theorem; Casorati-Weierstrass theorem Notes
    14 Evaluation of integrals by residue calculus
    15 MINI MIDTERM
    16 Theory of the gamma function Notes
    17 More about gamma; some examples Notes
    18 NO LECTURE - PENN STATE STUDY DAY
    19 Multiple-valued functions
    20 Computations with multiple-valued functions
    21 Conformal equivalence; conformal automorphisms of the plane; Schwarz' Lemma Notes
    22 Conformal automorphisms of the disk; hyperbolic geometry Notes
    23 Harmonic functions, maximum principle, Dirichlet problem, Poisson integral formula
    24 Equicontinuity, normal families, Harnack's principle
    25 Solution of the Dirichlet Problem a la Perron
    26 Analytic continuation
    27 The elliptic modular function j
    28 Picard's Theorem



     

    Functional Analysis, 14 lectures


    Textbook: N. Young, An Introduction to Hilbert Space, (CUP)
     
     
    1 Normed vector spaces, Banach spaces, Hilbert spaces Young 1,2
    2 Examples of Banach and Hilbert spaces Young 1,2
    3 Examples of Banach and Hilbert spaces Young 1,2
    4 The projection theorem, closed subspaces, dual of a Hilbert space Young 3,6
    5 Orthogonal decompositions; Bessels inequality Young 4,5
    6 Complete orthonormal sets; examples Young 4,5
    7 Operators on Hilbert space, adjoints, C*-algebras Young 7
    8 Basic spectral theory; the spectral radius formula. Young 7
    9 Abelian C*-algebras
    10 Compact operators.  Compact operators as limits of finite rank operators. Examples: integral operators Young 8
    11 Spectral theorem for compact self-adjoint operators. Young 8
    12 Sturm-Liouville theory Young 9-11
    13 The general spectral theorem for self-adjoint operators
    14 Review