After completing his famous
Foundations of Analysis (See AMS Chelsea Publishing,
Volume 79.H for the English Edition and AMS Chelsea Publishing,
Volume 141 for the German Edition, Grundlagen der
Analysis), Landau turned his attention to this book on
calculus. The approach is that of an unrepentant analyst, with
an emphasis on functions rather than on geometric or physical
applications. The book is another example of Landau's formidable
skill as an expositor. It is a masterpiece of rigor and
clarity.

Reviews

"And what a book it is! The marks of Landau's thoroughness and
elegance, and of his undoubted authority, impress themselves on
the reader at every turn, from the opening of the preface ... to
the closing of the final chapter. It is a book that all analysts
... should possess ... to see how a master of his craft like
Landau presented the calculus when he was at the height of his
power and reputation."

-- Mathematical
Gazette

Table of Contents

• Limits as n=infinity


• Logarithms, powers, and roots


• Functions and continuity


• Limits as $x=\xi$


• Definition of the derivative


• General theorems on the formation of the derivative


• Increase, decrease, maximum, minimum


• General properties of continuous functions on closed
  intervals


• Rolle's theorem and the theorem of the mean


• Derivatives of higher order; Taylor's theorem


• "0/0" and similar matters


• Infinite series


• Uniform convergence


• Power series


• Exponential series and binomial series


• The trigonometric functions


• Functions of two variables and partial derivatives


• Inverse functions and implicit functions


• The inverse trigonometric functions


• Some necessary algebraic theorems

• Definition of the integral


• Basic formulas of the integral calculus


• The integration of rational functions


• The integration of certain non-rational functions


• Concept of the definite integral


• Theorems on the definite integral


• The integration of infinite series


• The improper integral


• The integral with infinite limits


• The gamma function


• Fourier series


• Index of definitions


• Subject index