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Last Handout

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Things we didn't reach.

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

$\displaystyle e^{i\pi}+1=0, $

where $ \pi$ is the ratio of the circumference of a circle to its diameter, $ e$ is the basis of the natural logarithms, $ i$ is the square root of $ -1$, and 0 and $ 1$ are, well, you know. Less beautiful but more significant (and equally weired) is de Moivre's formula

$\displaystyle e^{ix} = \cos x + i\sin x. $

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

Hint. It helps to know that

$\displaystyle \displaystyle\sum_{n=1}^\infty F_nx^n=\frac{x}{1-x-x^2}. $

3. Bessel's function $ J_0$ is the solution of the differential equation $ x^2J_0''+xJ_0'+x^2J_0=0$ with $ J_0(0)=1$ and $ J_0'(0)=0$. A formula for $ J_0$ is

$\displaystyle J_0(x)=\sum_{k=0}^\infty\frac{(-1)^kx^{2k}}{(2^kk!)^2}. $

A few things we really have to say.

Definition. We say that a sequence of functions $ f_n$ converges to a function $ f$ uniformly on an interval $ I$ if for every $ \epsilon>0$ there is some $ N$ so that whenever $ n>N$ and $ x\in I$, we have that $ \vert f_n(x)-f(x)\vert<\epsilon$. Likewise we define uniform convergence for series $ \sum_{n=1}^\infty f_n(x)$.

Theorem 1. If $ f=\sum f_n$ uniformly on $ I$ and if for every $ n$ the function $ f_n$ is continuous on $ I$, then $ f$ is also continuous on $ I$.

Theorem 2. If $ f=\sum f_n$ uniformly on $ I$ and if for every $ n$ the function $ f_n$ is integrable on $ I$ then $ f$ is integrable on $ I$ and $ \int_I f=\sum\int_If_n$.

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

Theorem 4. If the series $ \sum a_nx^n$ is convergent for some $ x=x_0$, then it is uniformly and absolutely convergent on $ [-(x_0-\epsilon), x_0-\epsilon]$, for every $ \epsilon>0$. Thus ``all the good things'' happen for functions such as $ J_0(x)$.

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.

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.

Good Luck!!

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