Further Reading

Books

  1. The Elements of Real Analysis, by Robert G. Bartle
  2. Analysis in Euclidean Space, by Kenneth Hoffmann
  3. Advanced Calculus, by R. Creighton Buck
  4. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, by John H. Hubbard and Barbara Burke Hubbard
  5. Linear Algebra, by Kenneth Hoffman and Ray Kunze
  6. Complex Variables and Applications, by Ruel Vance Churchill and James Ward Brown
  7. Complex Analysis, by L. Ahlfors
  8. Theory of Functions of a Complex Variable (3 vols.), by A. I. Markushevich
  9. Visual Complex Analysis, by Tristan Needham
  10. Men of Mathematics, by E. T. Bell
  11. A History of Analysis, edited by Hans Neils Jahnke
  12. A History of Vector Calculus: The Evolution of the Idea of a Vectorial System, by Michael J. Crowe
  13. All the Mathematics You Missed: But Need to Know for Graduate School, by Thomas A. Garrity
  14. Mathematics: Form and Function, by Saunders MacLane
  15. Pi in the Sky: Counting, Thinking, and Being, by John D. Barrow

  16. Walter Rudin's Principles of Mathematical Analysis is a more advanced book on real analysis. It covers the ideas of limits, continuity, differentiation, and integration with more advanced results and less explanation. Rudin does not, however, cover complex analysis. The real analysis is done in the abstract perspective of metric spaces. After preparation with Real and Complex Analysis, you will be prepared to study baby Rudin.

  17. Walter Rudin's Real and Complex Analysis is at a higher level. Rudin immediately begins with the abstract Lebesgue integral, the most important and an extremely powerful integration process. The only prerequisites, however, are a knowledge of basic real analysis that you will mostly have gained from Real and Complex Analysis (not Rudin's book). He does use the idea of a metric space, which you will need to study from another source, like Rudin's first book above.

We also encourage you to read the following mathematical articles, which are all relevant to analysis. You can find them through JSTOR or at your college library. (Drew students can login here to JSTOR.)

Articles

  1. "The history of the calculus." Carl B. Boyer, The Two-Year College Mathematics Journal, Vol. 1, No. 1. (Spring, 1970), pp. 60-86. Has historical detail on calculus and analysis.
  2. "Everywhere differentiable, nowhere monotone functions" Y. Katznelson; Karl Stromberg, The American Mathematical Monthly, Vol. 81, No. 4. (Apr., 1974), pp. 349-354.
    This article constructs an elementary example of a function everywhere differentiable but monotone on no interval. It's hard enough to imagine a continuous function monotone on no interval. (For discontinuous functions, however, it is very easy--consider the function equal to 1 for rational points and 0 everywhere else.)
  3. "Another note on epsilons and deltas." Larry F. Bennett, The Two-Year College Mathematics Journal, Vol. 7, No. 3. (Sep., 1976), p. 18.
    This article provides a quick, delightful example of using the direct definition of the limit--the epsilons and deltas--to prove that the limit of xn as x approaches a is an. That is, no theorems on limits are used.
  4. "Newman's Short Proof of the Prime Number Theorem." D. Zagier, The American Mathematical Monthly, Vol. 104, No. 8. (Oct., 1997), pp. 705-708.
    The Prime Number Theorem is a spectacular application of complex function theory to the Riemann-zeta function. Readers looking for a challenge will enjoy this lively article.

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