 Leonard Euler April 15, 1707 - September 18, 1783 |
 Augustin-Louis Cauchy August 21, 1789 - May 23, 1857 |
 Karl Weierstrass October 31, 1815 - February 19, 1897 |
 Georg Friedrich Bernhard Riemann September 17, 1826 - July 20, 1866 |
Leonard Euler April 15, 1707 - September 18, 1783 |
Augustin-Louis Cauchy August 21, 1789 - May 23, 1857 |
Karl Weierstrass October 31, 1815 - February 19, 1897 |
Georg Friedrich Bernhard Riemann September 17, 1826 - July 20, 1866 |
COURSE: MATH 4227/5227
TIME AND PLACE: 11:15-12:25 TR in Brown 476
INSTRUCTOR: Dr. Robert Gardner OFFICE HOURS: Tesday 9:00-10:00 and by appointment.
OFFICE: Room 308F of Gilbreath Hall
PHONE: 439-6979 (308F Gilbreath), Math Department Office 439-4349
E-MAIL:
gardnerr@etsu.edu
WEBPAGE:
www.etsu.edu/math/gardner/gardner.htm (see my webpage for a copy of this course syllabus and updates for the course).
COURSE WEBPAGE: http://www.etsu.edu/math/gardner/4217/silspr07.htm
TEXT: An Introduction to Analysis, 2nd edition, by J. R. Kirkwood, Published by PWS Publishing Company and Waveland Press, Inc. 1995.
GRADING: Homework will be assigned and collected regularly. This will be the basis of your grade. Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate. Undergraduates (those registered for MATH 4227) will be required to do 75% of that required of graduates (those registered for MATH 5227).
A NOTE ABOUT HOMEWORK: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. Some of the homework problems are fairly standard for this class, and you may find proofs online. However, the online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class.
NOTE. This term, we will have to cover material faster than in Analysis 1. We will finish chapters 4 and 5 rather quickly. We will then get into the meat of the class and study integration, and sequences and series of functions. We will prove the most difficult result in the book: the Riemann-Lebesgue Theorem. This theorem gives necessary and sufficient conditions for a function to be Riemann integrable. The topic of sequences of functions and their interaction with integration is one of the main topics of real analysis. Our main result along these lines is Corollary 8-10(b). This result is a real foreshadowing of what you would see in a graduate level real analysis course.
OUTLINE: We will cover the following chapters and topics:
IMPORTANT DATES:
April 15, 2007 marks the 300th aniversary of the birth of one of the founders of modern analysis, Leonard Euler. Euler is responsable for some of the mathematical notation that, today, we take for granted: the notation f(x) for a function (1734), e for the base of natural logs" (1727), i for the square root of -1 (1777), π for pi, Σ for summation (1755), and Δ for finite differences (see the MacTutor History biography for details). In 1748, Euler published Introductio in analysin infinitorum, a text Carl Boyer calls "the foremost textbook of modern times." This work gave the foundations of what would become the modern analysis which we study in this class (with subsequent details given by Cauchy and Weierstrass). For details on Euler Tercentenary celebrations, click on the icon below:
In commemoration of our proof of the so called Riemann-Lebesgue Theorem ("A bounded function f is Riemann integrable on [
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