Principles of Analysis II

• Text: Principles of Mathematical Analysis
• Author: Walter Rudin
• ISBN: 0-07-054235-X
  This book will also be used for the second semester
• Publisher: McGraw-Hill

The second semester of Principles of Analysis focuses on applications of the Riemann-Stieltjes Integral analysis.

Look up your grades

1. Install Mozilla from Page for obtaining the Mozilla browser
   It will also be necessary to install math fonts for this browser. Information on how to install these can be found on
   The MathML Fonts Page
   For linux users, the necessary fonts can be found in: mozilla-math.tar.gz (these should be installed in X windows).

Note: Other web pages for this course will not necessarily even show up in other browsers! Once you have installed Mozilla and the fonts, you can check your installation by viewing:

The MathML Start Page

Use this to check whether MathML is working properly on your web browser. Unfortunately, the Windows version of Mozilla seems to have a bug that will cause it to complain about missing fonts even though the fonts are there (and Mozilla is using them correctly).

The term Analysis refers to advanced calculus as it was developed in the 19th and 20th centuries. It involves several broad topics including: the Lebesgue Integral, Complex Analysis and Functional Analysis. The methods and results of this work are needed for advanced work in fields as diverse as Topology, Quantum Mechanics and Ergodic Theory.

Topics covered

1. Chapter 7. Sequences and Series of functions
   • Uniform convergence and continuity
   • Stone-Weierstrass Theorem
   • Local Cauchy Theorem
2. Chapter 8. Some special functions
   • Power series
   • The Exponential and logarithmic functions
   • Trigonometric functions
   • Algebraic completeness of the complex field
   • Fourier series
   • Gamma function
3. Chapter 9. Functions of several variables
   • Linear Transformations
   • Differentiation
   • The Contraction Principle
   • The inverse function theorem
   • The implicit function theorem
   • The Rank theorem
   • Determinants
   • Derivatives of higher orders
   • Differentiation of integrals
4. Integration of differential forms
   • Primitive mappings
   • Partitions of Unity
   • Change of variables
   • Differential forms
   • Simplexes and chains
   • Stokes's theorem
   • Closed forms and exact forms
   • Volume elements: Green's theorem
5. Lebesgue Theory
   • Construction of Lebesgue Measure
   • Measure spaces
   • Simple functions
   • The Integral
   • Lebesgue's monotone convergence theorem
   • Lebesgue's dominated convergence theorem

Links

1. Complex Numbers home page
2. Complex Analysis Page
3. Graphics for complex analysis

Homework

1. p. 138 # 4, 8; p. 139 # 9, 10; p. 140 #12; p. 141. # 16 Due 4/7/2005
2. p. 165 # 4, p. 166 #6 p. 167 # 10 Due 4/14/2005
3. p. 168 #14, 15, 21, 23
4. p. 197 # 5, 6, 9, 10; p. 199 15, p. 200 19
5. Due 5/12/2005
   1. Prove, using Stirling's Formula that $$\underset{x\to \mathrm{\infty }}{lim}\frac{\Gamma (x+c)}{x^c\Gamma \left(x\right)}=1$$
   2. Show that: $$\underset{n\to \mathrm{\infty }}{lim}\sqrt{n}{\int }_{-1}^{+1}(1-x^2{)}^n\mathrm{dx}=\sqrt{\pi }$$ This is a strengthing of the result proved in the proof of Theorem 7.26.
   3. p. 99 #7; p. 239 #6
6. p. 240 14, 15; p. 241 16, 17; p. 242 27, 28
7. p. 292 20, 21; p. 333 #6,

Grades will be based on graded homework assignments given each week (mostly taken from the textbook).

Lecture Notes

Note: you may have to zoom on these images to see them clearly

1. 3-31-2005
2. 4-7-2005
3. 4-14-2005
4. 4-21-2005
5. 4-28-2005
6. 5-5-2005
7. 5-12-2005
8. 5-19-2005
9. 5-26-2005