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= Welcome to MWiki =
 
= Welcome to MWiki =
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== Theorem of the month ==
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== Theorems of the month ==
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=== Green's theorem ===
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=== Definition ===
  
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Given neighbourhood relations <math>B \subseteq {D}^{2}</math> for some <math>h</math>-domain <math>D \subseteq {}^{(\omega)}\mathbb{R}^{2}</math>, infinitesimal <math>h = |dBx|= |dBy| = |\curvearrowright B \gamma(t) - \gamma(t)| = \mathcal{O}({\hat{\omega}}^{m})</math>, sufficiently large <math>m \in \mathbb{N}^{*}, (x, y) \in D, {D}^{-} := \{(x, y) \in D : (x + h, y + h) \in D\}</math>, and a simply closed path <math>\gamma: [a, b[\rightarrow \partial D</math> followed anticlockwise, choosing <math>\curvearrowright B \gamma(t) = \gamma(\curvearrowright A t)</math> for <math>t \in [a, b[, A \subseteq {[a, b]}^{2}</math>, the following equation holds for sufficiently <math>\alpha</math>-continuous functions <math>u, v: D \rightarrow \mathbb{R}</math> with not necessarily continuous partial derivatives <math>\partial Bu/\partial Bx, \partial Bu/\partial By, \partial Bv/\partial Bx</math> and <math>\partial Bv/\partial By</math>:<div style="text-align:center;"><math>\int\limits_{\gamma }{(u\,dBx+v\,dBy)}=\int\limits_{(x,y)\in {{D}^{-}}}{\left( \frac{\partial Bv}{\partial Bx}-\frac{\partial Bu}{\partial By} \right)dB(x,y)}.</math></div>
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Let <math>f_n^*(z) = f(\eta_nz)</math> <em>sisters</em> of the Taylor series <math>f(z) \in \mathcal{O}(\mathbb{D})</math> centred on 0 on the domain <math>\mathbb{D} \subseteq {}^{\omega}\mathbb{C}</math> where <math>m, n \in {}^{\omega}\mathbb{N}^{*}</math> and <math>\eta_n^m := \underline{1}^{2^{\lceil m/n \rceil}}</math>. Then let <math>\delta_n^*f = \tilde{2}(f - f_n^*)</math> the <em>halved sister distances</em> of <math>f.</math> For <math>\mu_n^m := m!n!/(m + n)!</math>, <math>\mu</math> and <math>\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>\triangle</math>
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=== Representation theorem for integrals ===
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The Taylor series (see below) <math>f(z) \in \mathcal{O}(\mathbb{D})</math> centred on 0 on <math>\mathbb{D} \subseteq {}^{\omega}\mathbb{C}</math> gives for <math>\grave{m}, n \in {}^{\omega}\mathbb{N}^*</math><div style="text-align:center;"><math>{\uparrow}_0^z...{\uparrow}_0^{\zeta_2}{f(\zeta_1){\downarrow}\zeta_1\;...\;{\downarrow}\zeta_n} = \widetilde{n!} f(z\mu_n) z^n.\square</math></div>
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=== Representation theorem for derivatives ===
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For <math>{}^{\widetilde{\nu}}\dot{\mathbb{C}} \subset \mathbb{D} \subseteq {}^{\omega}\mathbb{C},</math> the Taylor series<div style="text-align:center;"><math>f(z):=f(0) + {\LARGE{\textbf{+}}}_{m=1}^{\omega }{\widetilde{m!}\,{{f}^{(m)}}(0){z^m}},</math></div><math>\varepsilon := \tilde{2}^j\tilde{r}, j \in {}^{\omega}\mathbb{Z}, n = \epsilon^{\sigma} \in {}^{\omega}\mathbb{N}^{*}, u :=\epsilon^{\tilde{n} \hat{\underline{\pi}}}</math> and <math>f</math>'s radius of convergence <math>r \in {}^{\nu}{\mathbb{R}}_{&gt;0}</math> imply<div style="text-align:center;"><math>{{f}^{(n)}}(0)=2^{jn}\acute{n}!{\LARGE{\textbf{+}}}_{k=1}^{n}{\delta_n^* f(\tilde{2}^j u^k)}.</math></div>
  
 
==== Proof: ====
 
==== Proof: ====
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Wlog the case <math>D := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : \partial D \rightarrow {}^{(\omega)}\mathbb{R}</math> is proved, since the proof is analogous for each case rotated by <math>\iota</math>, and every <math>h</math>-domain is a union of such sets. It is simply shown that<div style="text-align:center;"><math>\int\limits_{\gamma }{u\,dBx}=-\int\limits_{(x,y)\in {{D}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}</math></div>since the other relation is given analogously. Since the regions of <math>\gamma</math> where <math>dBx = 0</math> do not contribute to the integral, for negligibly small <math>t := h(u(s, g(s)) - u(r, g(r)))</math>, it holds that<div style="text-align:center;"><math>-\int\limits_{\gamma }{u\,dBx}-t=\int\limits_{r}^{s}{u(x,g(x))dBx}-\int\limits_{r}^{s}{u(x,f(x))dBx}=\int\limits_{r}^{s}{\int\limits_{f(x)}^{g(x)}{\frac{\partial Bu}{\partial By}}dBydBx}=\int\limits_{(x,y)\in {{D}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}.\square</math></div>
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Taylor's theorem<ref name="Remmert">[[w:Reinhold Remmert|<span class="wikipedia">Remmert, Reinhold</span>]]: ''Funktionentheorie 1'' : 3., verb. Aufl.; 1992; Springer; Berlin; ISBN 9783540552338, S. 165 f.</ref> and the properties of the roots of unity.<math>\square</math>
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== Reference ==
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<references />
  
 
== Recommended reading ==
 
== Recommended reading ==

Latest revision as of 23:05, 31 March 2024

Welcome to MWiki

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}(\mathbb{D}) }[/math] centred on 0 on the domain [math]\displaystyle{ \mathbb{D} \subseteq {}^{\omega}\mathbb{C} }[/math] where [math]\displaystyle{ m, n \in {}^{\omega}\mathbb{N}^{*} }[/math] and [math]\displaystyle{ \eta_n^m := \underline{1}^{2^{\lceil m/n \rceil}} }[/math]. Then let [math]\displaystyle{ \delta_n^*f = \tilde{2}(f - f_n^*) }[/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]

Representation theorem for integrals

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

[math]\displaystyle{ {\uparrow}_0^z...{\uparrow}_0^{\zeta_2}{f(\zeta_1){\downarrow}\zeta_1\;...\;{\downarrow}\zeta_n} = \widetilde{n!} f(z\mu_n) z^n.\square }[/math]

Representation theorem for derivatives

For [math]\displaystyle{ {}^{\widetilde{\nu}}\dot{\mathbb{C}} \subset \mathbb{D} \subseteq {}^{\omega}\mathbb{C}, }[/math] the Taylor series

[math]\displaystyle{ f(z):=f(0) + {\LARGE{\textbf{+}}}_{m=1}^{\omega }{\widetilde{m!}\,{{f}^{(m)}}(0){z^m}}, }[/math]

[math]\displaystyle{ \varepsilon := \tilde{2}^j\tilde{r}, j \in {}^{\omega}\mathbb{Z}, n = \epsilon^{\sigma} \in {}^{\omega}\mathbb{N}^{*}, u :=\epsilon^{\tilde{n} \hat{\underline{\pi}}} }[/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)=2^{jn}\acute{n}!{\LARGE{\textbf{+}}}_{k=1}^{n}{\delta_n^* f(\tilde{2}^j u^k)}. }[/math]

Proof:

Taylor's theorem[1] and the properties of the roots of unity.[math]\displaystyle{ \square }[/math]

Reference

  1. ↑ Remmert, Reinhold: Funktionentheorie 1 : 3., verb. Aufl.; 1992; Springer; Berlin; ISBN 9783540552338, S. 165 f.

Recommended reading

Nonstandard Mathematics

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