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= Welcome to MWiki =
 
= Welcome to MWiki =
 
== Theorem of the month ==
 
== Theorem of the month ==
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=== Finite representation for odd <math>\zeta</math>-arguments ===
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=== Green's theorem ===
  
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Using the digamma function <math>\psi</math>, it holds for <math>n \in {}^{\omega}2\mathbb{N}^{*}</math>, small <math>\varepsilon \in ]0, 1]</math> and <math>{{d}_{\varepsilon k n}}:={{\varepsilon}^{{\hat{n}}}}{e}^{\hat{n}2k\pi i}</math> that<div style="text-align:center;"><math>\zeta(\grave{n}) = \underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{-\varepsilon n}\sum\limits_{k=1}^{n}{\left( \gamma +\psi ({{d}_{\varepsilon k n}}) \right)}+\mathcal{O}(\varepsilon )</math></div>and<div style="text-align:center;"><math>\zeta(\grave{n}) = \underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{2\varepsilon n}\sum\limits_{k=1}^{n}{\left( \psi ({{d}_{\varepsilon k n}}{{i}^{\hat{n}2}})-\psi ({{d}_{\varepsilon k n}}) \right)}+\mathcal{O}({{\varepsilon }^{2}}).</math></div>
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Given neighbourhood relations <math>B \subseteq {A}^{2}</math> for some simply connected <math>h</math>-set <math>A \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 A, {A}^{-} := \{(x, y) \in A : (x + h, y + h) \in A\}</math>, and a simply closed path <math>\gamma: [a, b[\rightarrow \partial A</math> followed anticlockwise, choosing <math>\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)</math> for <math>t \in [a, b[, D \subseteq {[a, b]}^{2}</math>, the following equation holds for sufficiently <math>\alpha</math>-continuous functions <math>u, v: A \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 {{A}^{-}}}{\left( \frac{\partial Bv}{\partial Bx}-\frac{\partial Bu}{\partial By} \right)dB(x,y)}.</math></div>
  
 
==== Proof: ====
 
==== Proof: ====
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The claim results easily via the geometric series from <div style="text-align:center;"><math>\psi (z)+\gamma +\hat{z}=\sum\limits_{m=1}^{\omega }{\left( \hat{m}-\widehat{m+z} \right)}=-\sum\limits_{m=1}^{\omega }{\zeta(\grave{m}){{(-z)}^{m}}}=z\sum\limits_{m=1}^{\omega }{\hat{m}\widehat{m+z}}.\square</math></div>
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Wlog the case <math>A := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : \partial A \rightarrow {}^{(\omega)}\mathbb{R}</math> is proved, since the proof is analogous for each case rotated by <math>\iota</math>, and every simply connected <math>h</math>-set 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 {{A}^{-}}}{\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 {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}.\square</math></div>
  
 
== Recommended reading ==
 
== Recommended reading ==

Revision as of 16:53, 3 April 2020

Welcome to MWiki

Theorem of the month

Green's theorem

Given neighbourhood relations [math]\displaystyle{ B \subseteq {A}^{2} }[/math] for some simply connected [math]\displaystyle{ h }[/math]-set [math]\displaystyle{ A \subseteq {}^{(\omega)}\mathbb{R}^{2} }[/math], infinitesimal [math]\displaystyle{ h = |dBx|= |dBy| = |\curvearrowright B \gamma(t) - \gamma(t)| = \mathcal{O}({\hat{\omega}}^{m}) }[/math], sufficiently large [math]\displaystyle{ m \in \mathbb{N}^{*}, (x, y) \in A, {A}^{-} := \{(x, y) \in A : (x + h, y + h) \in A\} }[/math], and a simply closed path [math]\displaystyle{ \gamma: [a, b[\rightarrow \partial A }[/math] followed anticlockwise, choosing [math]\displaystyle{ \curvearrowright B \gamma(t) = \gamma(\curvearrowright D t) }[/math] for [math]\displaystyle{ t \in [a, b[, D \subseteq {[a, b]}^{2} }[/math], the following equation holds for sufficiently [math]\displaystyle{ \alpha }[/math]-continuous functions [math]\displaystyle{ u, v: A \rightarrow \mathbb{R} }[/math] with not necessarily continuous partial derivatives [math]\displaystyle{ \partial Bu/\partial Bx, \partial Bu/\partial By, \partial Bv/\partial Bx }[/math] and [math]\displaystyle{ \partial Bv/\partial By }[/math]:

[math]\displaystyle{ \int\limits_{\gamma }{(u\,dBx+v\,dBy)}=\int\limits_{(x,y)\in {{A}^{-}}}{\left( \frac{\partial Bv}{\partial Bx}-\frac{\partial Bu}{\partial By} \right)dB(x,y)}. }[/math]

Proof:

Wlog the case [math]\displaystyle{ A := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : \partial A \rightarrow {}^{(\omega)}\mathbb{R} }[/math] is proved, since the proof is analogous for each case rotated by [math]\displaystyle{ \iota }[/math], and every simply connected [math]\displaystyle{ h }[/math]-set is a union of such sets. It is simply shown that

[math]\displaystyle{ \int\limits_{\gamma }{u\,dBx}=-\int\limits_{(x,y)\in {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)} }[/math]

since the other relation is given analogously. Since the regions of [math]\displaystyle{ \gamma }[/math] where [math]\displaystyle{ dBx = 0 }[/math] do not contribute to the integral, for negligibly small [math]\displaystyle{ t := h(u(s, g(s)) - u(r, g(r))) }[/math], it holds that

[math]\displaystyle{ -\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 {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}.\square }[/math]

Recommended reading

Nonstandard Mathematics

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