Undergraduate Analysis
- 2nd ed.
- United States of America -- Springer -- 2010
- xv, 642p.
Part One - Review of Calculus.
0 Sets and Mappings.
I Real Numbers.
II Limits and Continuous Functions.
III Differentiation.
IV Elementary Functions.
V The Elementary Real Integral.
Part Two - Convergence.
VI Normed Vector Spaces.
VII Limits.
VIII Compactness.
IX Series.
X The Integral in One Variable.
Part Three - Applications of the Integral.
XI Approximation with Convolutions.
XII Fourier Series.
XIII Improper Integrals.
XIV The Fourier Integral.
Part Four - Calculus in Vector Spaces.
XV Functions on n-Space.
XVI The Winding Number and Global Potential Functions.
XVII Derivatives in Vector Spaces.
XVIII Inverse Mapping Theorem.
XIX Ordinary Differential Equations.
Part Five - Multiple Integration.
XX Multiple Integrals.
XXI Differential Forms.
Topics discussed include the classical test for convergence of series, Fourier series, polynomial approximation, the Poisson kernel, the construction of harmonic functions on the disc, ordinary differential equation, curve integrals, derivatives in vector spaces, multiple integrals, and others.