Some math notes

Some math notes

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Mittag-Leffler expansions
============

Let $f(z)$ be a complex function with an infinite number of *simple* poles $z_n$, and holomorphic at the origin. Suppose also that the poles are ordered in the following way
$$ 0 < | z_1| \leq | z_2 | \le \dots \le | z_n| \le \dots $$
If
1. $\mathcal{C}_n$ is a contour encircling all poles up to $z_n$,
2. $f$ is uniformly bounded on $\mathcal C_n$:
	$$ \exists A \in \mathbb{R} \,\, | \quad \forall (n \in \mathbb{N}, \, z \in \mathcal C_n ), \,\, | f(z)| < A. $$
Then $f$ admits the following Mittag-Leffler expansion:
$$ f(x) = f(0) + \sum_{n=1}^\infty \alpha_n \left( \frac{1}{z-z_n} + \frac{1}{z_n} \right). $$


Examples of Mittag-Leffler expansions
--------------------------

- $$ \csc(z) \equiv \frac{1}{\sin(z)}
		= \frac{1}{z} + \sum_{n \neq 0} (-1)^n \left( \frac{1}{z-k\pi} + \frac{1}{k\pi} \right)
		= \frac{1}{z}+ \sum_{n=1}^\infty (-1)^n \frac{2z}{z^2 - n^2 \pi^2}$$
- $$ \cot(z) \equiv \frac{\cos (z)}{\sin (z)}
		= \frac{1}{z} + \sum_{n \neq 0} \left( \frac{1}{z-n\pi} + \frac{1}{n\pi} \right)
		= \frac{1}{z} + \sum_{n=1}^\infty \frac{2z}{z^2 - n^2\pi^2}$$
- $$ \frac{1}{z \sin(z)}
		= \frac{1}{z^2} + \sum_{n \neq 0}  \frac{(-1)^n}{\pi n(z-\pi n)}
		= \frac{1}{z^2} + \sum_{n=1}^\infty \frac{(-1)^n}{n\pi} \frac{2z}{z^2 - \pi^2 n^2}$$
- $$ \sum_{n \neq 0} (-1)^n g(n) = - \pi \sum_i  \operatorname{Res}_{z=z_i}  \frac{g(z)}{\sin(\pi z)} $$
- $$ \sum_{n \neq 0} g(n) = - \pi \sum_i  \operatorname{Res}_{z=z_i}  g(z) \cot(\pi z) $$

Series summed with the Sommerfeld-Watson method:


Infinite products
=============


- $$ \tag 1 \prod_{n \neq 0,k} \frac{n-k}{n} = (-1)^{k+1}, \quad k \in \mathbb{Z}. $$
	- Proof:
$$ \prod_{n \neq 0,k} \frac{n-k}{n}
= \lim_{N \to \infty} \frac
	{(-N-k)(-N-k+1)\cdots(-k-1)(-k+1)\cdots(-1)(+1)\cdots(N-k)}
	{-N(-N+1)\cdots(-1)(+1)\cdots(k-1)(k+1)\cdots N}
\\ = \frac{(-1)^{N+k}}{-k} \lim_{N \to \infty}  \left( \frac
{(N+k)! (N-k)!}
	{-N(-N+1)\cdots(-1)(+1)\cdots(k-1)(k+1)\cdots N} \right)
\\ = \frac{(-1)^{N+k}}{-k} k(-1)^N \lim_{N \to \infty} \frac
	{(N+k)! (N-k)!}
	{(N!)^2}
= (-1)^{k+1} $$
	- math.SE questions:
		- this same proof: http://math.stackexchange.com/a/1143361/173147
		- slightly different proof: http://math.stackexchange.com/a/1143205/173147

-  $$\prod_{{n\geq1,\, n\neq k}}	\left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}, \quad k \in \mathbb{Z}. $$
	- math.SE question: http://math.stackexchange.com/q/1142703/173147

Weierstrass product expansion
-------------

Consider the Mittag-Leffler pole expansion of a meromorphic function $g(z)$:
$$ g(z) = g(0) + \sum_n \alpha_n \left( \frac{1}{z-z_n} + \frac{1}{z_n} \right). $$
Taking now $g(z) = f'(z)/f(z) \equiv \partial_L f(z)$ leads to
$$ \frac{f'(z)}{f(z)} = \frac{f'(0)}{f(0)} + \sum_n \alpha_n \left( \frac{1}{z-z_n} + \frac{1}{z_n} \right), $$
which integrating from $z=0$ to $z$ and taking the exponential gives
$$ f(z)=f(0) e^{\frac{z f'(0)}{f(0)}} \prod_n e^{ \alpha_n z/z_n} \left( 1 - \frac{z}{z_n} \right), $$
where $\alpha_n$ is the order of the $n$-th zero of $f$.

###Examples of product expansions:

- $$ \sin(z)
		= z \prod_{n \neq 0} e^{\frac{z}{n\pi}} \left( 1 - \frac{z}{n\pi} \right)
		= z \prod_{n=1}^\infty \left( 1 - \frac{z^2}{n^2\pi^2} \right)$$
- $$ \cos(z)
		= \prod_{n \in \mathbb{Z}} e^{\frac{2z}{(2n+1)\pi}} \left( 1 - \frac{2z}{\pi(2n+1)} \right)
	= \prod_{n=1}^\infty \left( 1 - \frac{4z^2}{(2n-1)^2\pi^2} \right) $$

Ordinary differential equations (ODEs)
==============

First order, homogeneous ODEs
------------
The general form of an homogeneous, first order, ordinary differential equation is:
$$ u'(x) + p(x)u(x) = 0, $$
with general solution (obtained for example integrating by parts):
$$ u(x) = A e^{-\int^x p}, $$
with $A$ a constant determined by the initial conditions.
More specifically, the solution of the Cauchy initial value problem
$$ \begin{cases} u' + p(x) u = 0 \\ u(x_0) = u_0 \end{cases} $$
is $$ u(x) = u(x_0) \exp \left( - \int_{x_0}^x p(t) dt \right) .$$

Second-Order Linear ODEs
--------

### Bessel's Equation
The Bessel's differential equation is the second-order ordinary differential equation given by
$$ x^2 y'' + x y' + (x^2-\alpha^2)y = 0, $$
where $\alpha$ is an arbitrary complex number, called the *order* of the Bessel's function.

- The canonical solutions of the Bessel's differential equation are the [Bessel's functions](https://en.wikipedia.org/wiki/Bessel_function).
- The most important cases are for $\alpha$ integer or half-integer. For $\alpha$ an integer we talk of **cylindrical Bessel's functions**. If $\alpha$ is half-integer we talk of **spherical Bessel's functions**.
- The Bessel's differential equation has a regular singularity at $z=0$ and an irregular singularity at $z=\infty$.
- From the indicial equation
$$ \rho^2 - n^2 = 0,$$
if $\rho=n$ the solution is
$$  y(z) = c_0 \, x^n \sum_{k=0}^\infty (-1)^k \frac{n!}{k!(n+k)!} \left( \frac{z}{2} \right)^{2k} $$
- f

Series Solution to ODEs (Frobenius' Method)
---------------
Resources on the Frobenius method:
- [Wikipedia article](https://en.wikipedia.org/wiki/Frobenius_method)

###Linear harmonic oscillator
Indicial equation: $ \rho (\rho-1) = 0$.
Papperitz equation: http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/papperitz.pdf

Other things
======

- [math.SE answer](http://math.stackexchange.com/a/1149667/173147): example of real-valued function infinitely differentiable at a point but *not* expressible as a power series at that point:
$$f(x) = \begin{cases} e^{-1/x^2}, & x \neq 0 \\
0, & x = 0 \end{cases}$$