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Theorems of the month

Definition

Let [math]\displaystyle{ f_n^*(z) = f(\eta_nz) }[/math] sisters of the Taylor series [math]\displaystyle{ f(z) \in \mathcal{O}(D) }[/math] centred on 0 on the domain [math]\displaystyle{ D \subseteq {}^{\omega}\mathbb{C} }[/math] where [math]\displaystyle{ m, n \in {}^{\omega}\mathbb{N}^{*} }[/math] and [math]\displaystyle{ \eta_n^m := i^{2^{\lceil m/n \rceil}} }[/math]. Then let [math]\displaystyle{ \delta_n^*f = (f - f_n^*)/2 }[/math] the halved sister distances of [math]\displaystyle{ f. }[/math] For [math]\displaystyle{ \mu_n^m := m!n!/(m + n)! }[/math], [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \eta }[/math] form an calculus, which can be resolved on the level of Taylor series and allows an easy and finite closed representation of integrals and derivatives.[math]\displaystyle{ \triangle }[/math]

Speedup theorem for integrals

The Taylor series (see below) [math]\displaystyle{ f(z) \in \mathcal{O}(D) }[/math] centred on 0 on [math]\displaystyle{ D \subseteq {}^{\omega}\mathbb{C} }[/math] gives for [math]\displaystyle{ \grave{m}, n \in {}^{\omega}\mathbb{N}^* }[/math]

[math]\displaystyle{ \int\limits_0^z...\int\limits_0^{\zeta_2}{f(\zeta_1)\text{d}\zeta_1\;...\;\text{d}\zeta_n} = \widehat{n!} f(z\mu_n) z^n.\square }[/math]

Speedup theorem for derivatives

For [math]\displaystyle{ \mathbb{B}_{\hat{\nu}}(0) \subset D \subseteq {}^{\omega}\mathbb{C}, }[/math] the Taylor series

[math]\displaystyle{ f(z):=f(0) + \sum\limits_{m=1}^{\omega }{\widehat{m!}\,{{f}^{(m)}}(0){z^m}}, }[/math]

[math]\displaystyle{ b_{\varepsilon n} := \hat{\varepsilon}\,\acute{n}! = 2^j, j, n \in {}^{\omega}\mathbb{N}^{*}, \varepsilon \in ]0, r^n[, {{d}_{\varepsilon k n}}:={{\varepsilon}^{{\hat{n}}}}{e}^{\hat{n}k\tau i} }[/math] and [math]\displaystyle{ f }[/math]'s radius of convergence [math]\displaystyle{ r \in {}^{\nu}{\mathbb{R}}_{>0} }[/math] imply

[math]\displaystyle{ {{f}^{(n)}}(0)=b_{\varepsilon n}\sum\limits_{k=1}^{n}{\delta_n^* f({{d}_{\varepsilon k n}})}. }[/math]

Proof:

Taylor's theorem and the properties of the roots of unity.[math]\displaystyle{ \square }[/math]

Recommended reading

Nonstandard Mathematics