$$
H_{C_1}(s) = \frac
{1}
{C_1s}
$$
$$
H_{R_1}(s) = \frac
{T_1s + 1}
{C_1s}
$$
$$
H_{C_2}(s) = \frac
{T_1s + 1}
{C_2T_1s^2 + (C_2 + C_1)s}
$$
$$
H_{R_2}(s) = \frac
{T_1T_2s^2 + (T_1 + T_2 + C_1R_2)s + 1}
{T_1C_2s^2 + (C_1 + C_2)s}
$$
$$
H_{C_3}(s) = \frac
{T_1T_2s^2 + (T_1 + T_2 + C_1R_2)s + 1}
{T_1T_2C_3s^3 + ((T_1 + T_2)C_3 + T_1C_2 + C_1R_2C_3)s^2 + (C_1 + C_2 + C_3)s}
$$
$$
H_{R_3}(s) = \frac
{T_1T_2T_3s^3 + ((T_1 + T_2)T_3 + T_1C_2R_3 + T_3C_1R_2 + T_1T_2)s^2 + (T_1 + T_2 + T_3 + R_3C_2 + R_2C_1 + R_3C_1)s + 1}
{T_1T_2C_3s^3 + ((T_1 + T_2)C_3 + T_1C_2 + C_1R_2C_3)s^2 + (C_1 + C_2 + C_3)s}
$$
$$
H_{C_4}(s) = \frac
{T_1T_2T_3s^3 + ((T_1 + T_2)T_3 + T_1C_2R_3 + T_3C_1R_2 + T_1T_2)s^2 + (T_1 + T_2 + T_3 + R_3C_2 + R_2C_1 + R_3C_1)s + 1}
{T_1T_2T_3C_4s^4 + ((T_1 + T_2)T_3C_4 + T_1T_2(C_3 + C_4) + T_1C_2R_3C_4 + T_3C_1R_2C_4)s^3 + ((T_1 + T_2)(C_3 + C4) + T_1C_2 + T_3C_4 + C_1R_2C_3 + R_3C_2C_4 + R_2C_1C_4 + R_3C_1C_4)s^2 + (C_1 + C_2 + C_3 + C_4)s}
$$
Wow, this is just great!
First time I was an online page that could print this big (i mean the last one) of a Latex expression. I'm impressed.
But, it is difficult to write. The preview pane flashes between Latex code and Latex images as I type my text. Try editing this message and see for yourself.
Matrices and Alignment
Simple Alignment
\$\begin{matrix}
11 & 12 & 13\\
21 & 22 & 23
\end{matrix}\$
\$\begin{matrix}
11 & 12 & 13\\
21 & 22 & 23
\end{matrix}\$
A Matrix
\$\begin{bmatrix}
11 & 12 & 13\\
21 & 22 & 23
\end{bmatrix}\$
\$\begin{bmatrix}
11 & 12 & 13\\
21 & 22 & 23
\end{bmatrix}\$
Some Other Matrix Notations
\$\begin{pmatrix}
11 & 12 & 13\\
21 & 22 & 23
\end{pmatrix}\$
\$\begin{pmatrix}
11 & 12 & 13\\
21 & 22 & 23
\end{pmatrix}\$
\$\begin{Bmatrix}
11 & 12 & 13\\
21 & 22 & 23
\end{Bmatrix}\$
\$\begin{Bmatrix}
11 & 12 & 13\\
21 & 22 & 23
\end{Bmatrix}\$
Norm
\$\begin{Vmatrix}
11 & 12 & 13\\
21 & 22 & 23
\end{Vmatrix}\$
\$\begin{Vmatrix}
11 & 12 & 13\\
21 & 22 & 23
\end{Vmatrix}\$
Partial Expressions
\$f(x) =Â
\left\{\begin{matrix}
x; & x \leq 0\\Â
0; & otherwise
\end{matrix}\right.\$
\$f(x) =Â
\left\{\begin{matrix}
x; & x \leq 0\\Â
0; & otherwise
\end{matrix}\right.\$
Multiple Line Alignment
\$\begin{align}
f(x, y, z) = & & x^3 & + & 2x^2 & + & x & + & 3 \\
& + & 4y^3 & & & + & 5x & - & 1 \\
& & & - & z^2 & - & 2z & + & 2
\end{align}\$
\$\begin{align}
f(x, y, z) = & & x^3 & + & 2x^2 & + & x & + & 3 \\
& + & 4y^3 & & & + & 5x & - & 1 \\
& & & - & z^2 & - & 2z & + & 2
\end{align}\$
Usage of Matrices and Alignment in Practice
$$H(s) = \frac{\sum_{i=0}^{n-1} b_is^i}{s^n + \sum_{i=0}^{n-1} a_is^i} = \frac{b_{n-1}s^{n-1} + b_{n-2}s^{n-2} + b_{n-3}s^{n-3} + \dots + b_0}
{s^n + a_{n-1}s^{n-1} + a_{n-2}s^{n-2} + \dots + a_0}\\
\ \\
\begin{matrix}
\mathbf{\dot{x}} & = & A\mathbf{x} + Bu(t)\\
y(t) & = & C\mathbf{x} + Du(t)
\end{matrix}\\
\ \\
\begin{matrix}
\begin{bmatrix}
\dot{x}_1\\
\dot{x}_2\\
\dot{x}_3\\
\vdots \\
\dot{x}_n
\end{bmatrix}
&
=
&
\begin{bmatrix}
-b_{n-1} & -b_{n-2} & -b_{n-3} & -b_{n-4} & \cdots & -b_0 \\
1 & 0 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \vdots & \cdots & 0 \\
0 & 0 & 1 & 0 & \cdots & 0 \\
\vdots & \vdots & 0 & \ddots & 0 & \vdots\\
0 & 0 & 0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2\\
x_3\\
\vdots \\
x_n
\end{bmatrix}
+
\begin{bmatrix}
1\\
0\\
0\\
\vdots \\
0
\end{bmatrix}
u(t)
\\
y(t)
&
=
&
\begin{bmatrix}
1 & 0 & 0 & \cdots & 0
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2\\
x_3\\
\vdots \\
x_n
\end{bmatrix}
\end{matrix}$$
Notice that I used nested matrices.
$$H(s) = \frac{\sum_{i=0}^{n-1} b_is^i}{s^n + \sum_{i=0}^{n-1} a_is^i} = \frac{b_{n-1}s^{n-1} + b_{n-2}s^{n-2} + b_{n-3}s^{n-3} + \dots + b_0}
{s^n + a_{n-1}s^{n-1} + a_{n-2}s^{n-2} + \dots + a_0}\\
\ \\
\begin{matrix}
\mathbf{\dot{x}} & = & A\mathbf{x} + Bu(t)\\
y(t) & = & C\mathbf{x} + Du(t)
\end{matrix}\\
\ \\
\begin{matrix}
\begin{bmatrix}
\dot{x}_1\\
\dot{x}_2\\
\dot{x}_3\\
\vdots \\
\dot{x}_n
\end{bmatrix}
&
=
&
\begin{bmatrix}
-b_{n-1} & -b_{n-2} & -b_{n-3} & -b_{n-4} & \cdots & -b_0 \\
1 & 0 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \vdots & \cdots & 0 \\
0 & 0 & 1 & 0 & \cdots & 0 \\
\vdots & \vdots & 0 & \ddots & 0 & \vdots\\
0 & 0 & 0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2\\
x_3\\
\vdots \\
x_n
\end{bmatrix}
+
\begin{bmatrix}
1\\
0\\
0\\
\vdots \\
0
\end{bmatrix}
u(t)
\\
y(t)
&
=
&
\begin{bmatrix}
1 & 0 & 0 & \cdots & 0
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2\\
x_3\\
\vdots \\
x_n
\end{bmatrix}
\end{matrix}$$
It crashes if you play with big expressions too much!

It crashed several times until I finish my post.
(My browser: Google Chrome 19.0.1084.56 m)
However, the good thing is, Stack Exchange fully and successfully recovers the lost message after every crash.
\$\Rightarrow \LaTeX\$
\$\Rightarrow \LaTeX\$ \$\endgroup\$ – Kevin Vermeer Mar 8 '11 at 22:20