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

- Each homework question will be graded out of 10 points. The best ten of your homework assignments will contribute to your total score, up to a maximum of 300 points.
- The final exam will consist of eight questions (five on complex analysis, three on Hilbert space) each of which will be graded out of 50 points. The best four questions will contribute to your total score, up to a maximum of 200 points.
- Grades will be assigned on the basis of your total score. I anticipate that around 410 points will suffice for an A grade, but this may be changed as the course progresses; a more detailed grading scale will be posted by midsemester. Please note that this cutoff is lower than you may be used to in an undergraduate course; this reflects the fact there is usually room for improvement in an extended-answer homework solution.
No late homework will be accepted under any circumstances.

- Homework 1, due September 12th
- Homework 2, due September 19th
- Homework 3, due September 26th
- Homework 4, due October 3rd
- Homework 5, due October 17th
- Homework 6, due October 27th
- Homework 7, due November 3rd
- Homework 8, due November 10th
- Homework 9, due November 17th (Solutions to this homework.)
- Homework 10, due November 24th
- Homework 11, due December 3rd
- Homework 12, due December 10th

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

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.

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 |

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 |