News: Homework 5 with solutions is available here.
           The comments/solutions to take-home midterm exam are now available.
           The final exam is now available.

These courses is what used to be an undegraduate analysis sequence 411-412 and has been modified to become 4111-4121. They cover the theory of one and several variable calculus followed by the introduction to Lebesgue theory.  This sequence is usually required for those wishing to complete the so-called "traditional" math major, a group normally limited to students contemplating going on to do graduate work in mathematics, physics, mathematical economics, or some other highly quantitative discipline in which theory plays a large role.  Students who are not traditional math majors are welcome to take the course provided they have an interest in understanding the theoretical basis for calculus. Those interested in building up their prowess with techniques for doing calculus problems and uninterested in understanding why the techniques work are STRONGLY ADVISED NOT to take 4111-21.  While it's true that occasionally techniques will be introduced in 4111-4121 which aren't covered in other calculus courses, these will be few and far between; also, most of the lecture time will be devoted to the theory of why these methods work rather than illustrations of the calculations.  In brief, there is virtually nothing in common between the style of earlier math courses (nothing proven, hand-waving justifications, reliance on physical intuition, working out numerical exercises, developing computational skills) and the style of 4111-21 (nearly everything proven with a fairly high degree of rigor, a large part of the homework and exam questions theoretical with only a few numerical problems tossed in for good measure, emphasis on building up skills at proving theorems and developing mathematical intuition).



Location:   Lopata Hall 301 -T-T--- 10:00AM 11:30AM

Instructor: Ilya Krishtal

             Office:  Cupples I, Room 202 (between the floors)
             Office Hours:  TBA or by appointment
             Office Tel:    935-6785 (has voice-mail)
             E-mail:        krishtal @

Textbooks: Bartle, Robert G. The elements of integration and Lebesgue measure.
           Containing a corrected reprint of the 1966 original [ The elements of
           integration, Wiley, New York; MR0200398 (34 \#293)]. Wiley Classics
           Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc.,
           New York, 1995. xii+179 pp. ISBN: 0-471-04222-6

           Rosenlicht, Maxwell. Introduction to analysis.
           Dover Publications, Inc., New York, 1986. viii+254 pp. ISBN 0-486-65038-3

Topics: Riemann integration; measurable functions; measures; the Lebesgue integral; integrable functions; L^p spaces; modes of convergence; decomposition of measures; product measures; Lebesgue measure. Other topics will be tossed in if time permits. Not all of the topics are covered in the textbooks. For those that are not other references or a handout will be provided.

Exams:  There will be one mid-term take home exam, approximately in the middle of the semester, and a final exam. While the mid-term exam will ask for proofs, definions, examples and counter-examples, and problems most of which were covered in class or analagous to homework problems, the final exam will consist entirely of problems not covered in class. We will settle on the exact dates when everyone has a firm schedule for exam week. 

Homework:  There will be weekly homework assignments to write up and hand in.  Usually the homework will be due on Thursday and the assignment handed out the previous Tuesday.  The homework will consist of a mixture of selected exercises from the textbook and supplementary exercises made up by the instructor. Click on current homework to get it.

Grading:   The mid-term exams will count 30% toward the final grade, the final exam and the final homework average will count 35% each. The actual cut-offs for the letter grades will be determined by the performance of the class. Cut-offs for 4111 may be used as a guide.

Academic Integrity: As with all Washington University courses, cheating on exams will be taken very seriously with evidence supporting a cheating allegation forwarded to the Arts and Sciences Integrity Committee for adjudication.  When the Committee concludes that a student cheated on an exam, it normally directs the instructor to assign the student a failing grade for the course.
      Cheating on homework consists of either blindly copying off someone else's solutions or not acknowledging the receipt of assistance from others in completing the assignment.  It's anticipated that students will make a genuine effort to solve the homework problems themselves.
However, if the effort has lead nowhere, discussing problems with others is a way to avoid frustration and gain useful insight. All students are expected to write up their own assignments and to indicate in a short note at the top of the first page the names of any people (other than the instructor) with whom they discussed the problems or from whom they received some hints.  Violation of these requests will result in an instructor-imposed penalty (e.g., something like half credit for the assignment) but won't be treated as a "hanging" offense--in particular, won't be brought to the attention of the Arts and Sciences Integrity Committee.

  Old 4111 News: You can download test 1 , test 2  and the FINAL TEST with my comments/solutions.
                             You are welcome
to contact me if you want me to elaborate on those.
Here is the graph of the function in problem 4, hw 10.
                             Here is the proof of the
Weierstrass approximation theorem.