Analysis II
Math 555, Spring 1998

Professor Robert Sharpley
sharpley@math.sc.edu
Department of Mathematics
University of South Carolina

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Table of Contents

(Math555 Info and Syllabus)

Lectures
(Html lectures developed during Spring 98. These typically state definitions, theorems, and examples, in a such a way that the steps to fill in the details are suitable for the student to perform.)

• (Special Functions)
• Metric Spaces
• (Introduction)
• (Open and Closed Sets)
• (Limits and Continuity)
• (Completeness)
• Compactness and its applications
• (Compactness and the Heine-Borel Theorem)
• (Total Boundedness, Separability, Countable Compactness)
• (Connectedness)
• Riemann-Stieltjes Integration
• (Introduction)
• (Conditions for Existence)
• (Properties of the Riemann- Stieltjes integral)
• (Additional Properties of the Riemann- Stieltjes integral)
• Infinite Series: both in the Scalars and Normed Linear Spaces
• (Introduction to infinite series)
• (Convergence Tests)
• (Ratio and n-th Root Convergence Tests)
• (lim sup  and  lim inf)
• Sequences and Series of Functions
• (Power Series and their Radius of Convergence)
• (Interchange of Limit Operations)
• Taylor's Approximation
• Exponential and Trigonmetric Functions in C
• Weierstrass Approximation Theorem
• Ascoli's Theorem
• Existence and Uniqueness of ODE
• Fourier Series and Solutions to Partial Differential Equations