# Symbols and Notation

:::{table} 
:align: left  
|  Symbol   | Description |
|-----------| -------------------------- |
| $\N$, $\Z^{+}$      | the set of natural (positive integer) numbers $\{1, 2, 3, \ldots\}$ |
| $\Z$      | the set of integer numbers $\{\ldots, -2, -1, 0, 1, 2, 3, \ldots\}$ |
| $\N_0$, $\Z^{+}_0$| the set of whole (non-negative integer) numbers $\{ 0, 1, 2, 3, \ldots\}$ |
| $\R$      | the set of real numbers|
| $\mathbf{\hat{i}}$      | the imaginary unit, $\mathbf{\hat{i}}^2=-1$ |
| $z$      | a complex number|
| $\|z\|$   | the modulus of $z$ |
| $\arg(z)$ | the argument of $z$ |
| $\Re(z)$, $\mathrm{Re}(z)$  | the real component of $z$|
| $\Im(z)$, $\mathrm{Im}(z)$  | the imaginary component of $z$|
| $\theta$  | angle |
| $\mathbb{C}$      | the set of complex numbers $\{z \| z=x+\mathbf{\hat{i}} y~, ~ x \in \R, y \in \R\}$ |
| $\{y_n\}_{n=0}^{+\infty}$ | a sequence defined on $\N_0$ |
| $\D$      | differential operator|
| $\Delta$  | forward difference operator|
| $\nabla$  | backward difference operator|
| $\E$      | forward shift operator |
| $\E^{-1}$ | backward shift operator|   
| $I$       | interval of definition |
| $t$       | time variable |
| $h$       | step size, step length |
| $k$       | number of steps |
| $\alpha_i$, $\beta_i$ | the coefficients of a linear multistep method ($i=0, 1, 2, \ldots, k$)|
| $\epsilon$ | error           |
| $T$, LTE    | local (truncation) error| 
| $\L(\D)$   | linear differential operation|
| $\L(\E)$   | linear shift operation|
| $\L(z)$    | characteristic polynomial|
| $\L(z, h\lambda)$, $\pi(z, h\lambda)$    | characteristic polynomial, stability polynomial|
| $\rho(z)$    | first characteristic polynomial|
| $\sigma(z)$    | second characteristic polynomial|
| $\begin{pmatrix} n \\ k \end{pmatrix}$, $^n C_r$        | binomial coefficient |
|  $\forall$    | for all |
|  $\exists$    | there exists |
|  $\implies$    | implies; if ... then |
| *iff*, $\iff$| if and only if, equivalent to |
| $\because$| because, since |
| $\therefore$| therefore |
| $\subset$| is a proper subset of |
| $\subseteq$| is a subset of |
| $\in$| is an element of |
| $\notin$| is not an element of |
| $\angle$| angle, argument |
| $i$, $j$, $k$, $m$, $n$, $p$    | whole numbers|
| $r$, $s$  | real numbers |
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