© | Dror Bar-Natan: Classes: 2004-05: Math 157 - Analysis I: | (111) |
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this document in PDF: LastHandout.pdf

**Things we didn't reach.**

**1. Euler's Formula** (aka ``the most beautiful formula in
mathematics''):

**2. Question. ** Find a formula for the 'th term in the
sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

**Hint. ** It helps to know that

**3. Bessel's function ** is the solution of the
differential equation
with and
. A formula for is

**A few things we really have to say.**

**Definition. ** We say that a sequence of functions
converges to a function *uniformly* on an interval if for every
there is some so that whenever and , we have
that
. Likewise we define *uniform convergence*
for series
.

**Theorem 1. ** If
uniformly on and if for
every the function is continuous on , then is also
continuous on .

**Theorem 2. ** If
uniformly on and if for
every the function is integrable on then is integrable on
and
.

**Theorem 3. ** A similar though slightly more complicated
statement holds for derivatives.

**Theorem 4. ** If the series
is convergent for
some , then it is uniformly and absolutely convergent on
, for every
. Thus ``all the
good things'' happen for functions such as .

**On the Final Exam.** It will take place, as dictated by
the Higher Authorities, on Wednesday May 4, 7-10PM (late!) at the
Upper Small Gymnasium, Benson Building, 320 Huron Street (across from
Sidney Smith, south of Harbord Street), Third Floor. The material is
very easy to define: *Everything*. In more details, this is
chapters 1-15, 18-20 and 22-24 of Spivak's book, minus appendices
plus the appendix to chapter 19 plus some extra material on convexity
(as it was discussed in class). If any question will relate to chapter 24,
it will be relatively simple and will not require knowing proofs.

**Preparing for the Final Exam. **

- Re-read your notes and make sure that you understand
*everything*. - Re-read the relevant chapters of Spivak's book
and make sure that you understand
*everything*. - You may want to prepare a list of all topics touched in class (you may reach 200 or even 400), and you may want to go over this list several times until you are sure you understand everything in full.
- Make sure that you can solve every homework problem assigned or recommended.
- Take a good look at exams, sample exams and exam solutions from previous years. (Scroll down to the bottom of this class' web site and find the relevant links).
- Come to my office hours on Monday and Tuesday May 2nd and 3rd, from 10AM until 1PM (or even later, if there's demand), at the Math Aid Centre, SS 1071.
- It is much more fun to work in a group!

An often-asked question is ``Do we need to know proofs?''. The answer is
**Absolutely**. Proofs are often the deepest form of understanding,
and hence they are largely what this class is about. The ones I show in
class are precisely those that I think are the most important ones,
thus they are the ones you **definitely** need to know.

**Last Comment. ** Please remember that absolutely all
homework is due Friday April 15, 2PM, at the
Math Aid Centre.

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Dror Bar-Natan 2005-04-06