|
|
| Line 24: |
Line 24: |
| | <math> | | <math> |
| | \begin{align} | | \begin{align} |
| − | W_n (s) | + | \label{def:Wns} W_n (s) |
| | &:= | | &:= |
| | \int_{[0, 1]^n} | | \int_{[0, 1]^n} |
| Line 30: |
Line 30: |
| | \end{align} | | \end{align} |
| | </math> | | </math> |
| | + | |
| | <math> | | <math> |
| | \begin{align} | | \begin{align} |
| | + | |
| | \newcommand{\Re}{\mathrm{Re}\,} | | \newcommand{\Re}{\mathrm{Re}\,} |
| | \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} | | \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} |
Revision as of 13:20, 21 September 2012
[math]
\begin{align}
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}
[/math]
[math]
\begin{align}
\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}
\end{align}
[/math]
[math]
\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}
[/math]
[math]
\begin{align}
\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
\label{eq:W3k}
W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
\end{align}
[/math]
