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Andreas Plank: Hilfe:Kategorisieren von Seiten: Kategorie: Wiki entfernt; Kategorie: Wiki/en hinzugefügt mittels HotCat-Helferlein (s. Benutzereinstellungen)

2018-06-28T22:32:43Z

Hilfe:Kategorisieren von Seiten: Kategorie: Wiki entfernt; Kategorie: Wiki/en hinzugefügt mittels HotCat-Helferlein (s. Benutzereinstellungen)

Tony Ford am 16. April 2013 um 16:01 Uhr

2013-04-16T16:01:43Z

Shure am 19. März 2013 um 12:12 Uhr

2013-03-19T12:12:13Z

Nikolayhg am 24. Mai 2012 um 22:02 Uhr

2012-05-24T22:02:42Z

Admin am 24. Mai 2012 um 21:10 Uhr

2012-05-24T21:10:54Z

Admin am 24. Mai 2012 um 21:01 Uhr

2012-05-24T21:01:13Z

Admin am 24. Mai 2012 um 20:51 Uhr

2012-05-24T20:51:16Z

Admin am 24. Mai 2012 um 20:10 Uhr

2012-05-24T20:10:25Z

Admin: Die Seite wurde neu angelegt: „ ===Integral Equation=== $\phi_n(\kappa) = \frac{1}{4\pi^2\kappa^2} \int_0^\infty \frac{\sin(\kappa R)}{\kappa R} \frac{\partial}{\partial R} \left[R^…“$

2012-05-24T20:05:43Z

Die Seite wurde neu angelegt: „ ===Integral Equation=== <math>\phi_n(\kappa) = \frac{1}{4\pi^2\kappa^2} \int_0^\infty \frac{\sin(\kappa R)}{\kappa R} \frac{\partial}{\partial R} \left[R^…“

Neue Seite

===Integral Equation===
<math>\phi_n(\kappa)
= \frac{1}{4\pi^2\kappa^2} \int_0^\infty \frac{\sin(\kappa R)}{\kappa R} \frac{\partial}{\partial R} \left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR</math>

← Abfassung vorher		Version vom 28. Juni 2018, 22:32 Uhr
Zeile 28:		Zeile 28:
	at the end of the paper.		at the end of the paper.

−	[[Category: Wiki]]	+	[[Category:Wiki/en]]

← Abfassung vorher		Version vom 16. April 2013, 16:01 Uhr
Zeile 28:		Zeile 28:
	at the end of the paper.		at the end of the paper.

−	[[Category: ~~Tests~~]]	+	[[Category: Wiki]]

@@ Zeile 22: / Zeile 22: @@
 \begin{align}
    \label{eq:W3k}
-   W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
+   W_3(k) &= \Re \, \pFq32{ \\ \frac12, -\frac k2, -\frac k2}{1, 1}{4}.
 \end{align}
 Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.

← Abfassung vorher		Version vom 24. Mai 2012, 22:02 Uhr
Zeile 27:		Zeile 27:
	The reason for \eqref{eq:W3k} was long a mystery, but it will be explained		The reason for \eqref{eq:W3k} was long a mystery, but it will be explained
	at the end of the paper.		at the end of the paper.
		+
		+	[[Category: Tests]]

← Abfassung vorher		Version vom 24. Mai 2012, 21:10 Uhr
Zeile 1:		Zeile 1:
−	<~~math~~>\~~text~~ {~~abcdefghijklmnopqrstuvwxyzàáâãäåæçčďèéěêëìíîïňñòóôõöřšť÷øùúůûüýÿž~~}~~</math~~	+	<!-- some LaTeX macros we want to use: -->
		+	$
		+	\newcommand{\Re}{\mathrm{Re}\,}
		+	\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg\| {#5} \right)}
		+	$
		+
		+	We consider, for various values of $s$, the $n$-dimensional integral
		+	\begin{align}
		+	\label{def:Wns}
		+	W_n (s)
		+	&:=
		+	\int_{[0, 1]^n}
		+	\left\| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right\|^s \mathrm{d}\boldsymbol{x}
		+	\end{align}
		+	which occurs in the theory of uniform random walk integrals in the plane,
		+	where at each step a unit-step is taken in a random direction. As such,
		+	the integral \eqref{def:Wns} expresses the $s$-th moment of the distance
		+	to the origin after $n$ steps.
		+
		+	By experimentation and some sketchy arguments we quickly conjectured and
		+	strongly believed that, for $k$ a nonnegative integer
		+	\begin{align}
		+	\label{eq:W3k}
		+	W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
		+	\end{align}
		+	Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.
		+	The reason for \eqref{eq:W3k} was long a mystery, but it will be explained
		+	at the end of the paper.

← Abfassung vorher		Version vom 24. Mai 2012, 21:01 Uhr
Zeile 1:		Zeile 1:
−	<math>\~~sqrt~~{~~\sqrt{\sqrt{x}}~~}</math>	+	<math>\text {abcdefghijklmnopqrstuvwxyzàáâãäåæçčďèéěêëìíîïňñòóôõöřšť÷øùúůûüýÿž}</math

← Abfassung vorher		Version vom 24. Mai 2012, 20:51 Uhr
Zeile 1:		Zeile 1:
−	<math>	+	<math>\sqrt{\sqrt{\sqrt{x}}}</math>
−	\~~operatorname~~{~~erfc}(x) =~~
−	\~~frac{2}~~{\sqrt{~~\pi}} \int_x^{\infty} e^{-t^2}\,dt =~~
−	~~\frac{e^{-~~x^2}~~}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n~~}}
−	</math>

@@ Zeile 1: / Zeile 1: @@
 <math>
-===Integral Equation===
+  \operatorname{erfc}(x) =
- <math>\phi_n(\kappa)
+  \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt =
- = \frac{1}{4\pi^2\kappa^2} \int_0^\infty \frac{\sin(\kappa R)}{\kappa R}  \frac{\partial}{\partial R}  \left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR</math>
+  \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}