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This wiki is a resource for time scale calculus. Time scale calculus is a unification and extension of differential and difference calculus in which one does calculus upon a set $\mathbb{T}$ of real numbers called a time scale. When $\mathbb{T}=\mathbb{R}$ the resulting theory becomes differential calculus, when $\mathbb{T}=\mathbb{Z}$ the resulting theory becomes difference calculus, and when $\mathbb{T}=\{1,q,q^2,\ldots\}, q>1$, the resulting theory becomes the $q$-calculus. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the Cantor set are also subsumed in the theory.

A result proven in time scale calculus implies the result for all choices of $\mathbb{T}$ so a result in time scale calculus immediately implies the result in differential calculus, the same result in difference calculus, the same result in $q$-calculus, the same result in calculus on the Cantor set, and countless others. For an example of this phenomenon, see the familiar properties of the $\Delta$-derivative to classical differentiation or to taking a forward difference.

See the Python library timescalecalculus on GitHub and its documentation.

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# Time scales calculus

## $\Delta$-special functions on time scales $\cos_p$ $\cosh_p$ $e_p$ $g_k$ $h_k$ $\sin_p$ $\sinh_p$

## Hilger complex plane and friends

$\Huge\mathbb{A}_h$
Hilger alternating axis
$\Huge\mathbb{I}_h$
Hilger circle
$\Huge\mathbb{C}_h$
Hilger complex plane
$\Huge\mathrm{Im}_h$
Hilger imaginary part
$\Huge\mathring{\iota}$
Hilger pure imaginary
$\Huge\mathbb{R}_h$
Hilger real axis
$\Huge\mathrm{Re}_h$
Hilger real part

## $\Delta$-Inequalities

 Bernoulli Bihari Cauchy-Schwarz Gronwall Hölder Jensen Lyapunov Markov Minkowski Opial Tschebycheff Wirtinger

## Examples of time scales

1. The real line: $\mathbb{R}$
2. The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
3. Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
4. Quantum numbers ($q>1$): $\overline{q^{\mathbb{Z}}}$
5. Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}$
6. Square integers: $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$
7. Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
8. The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
9. Isolated points: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$