User:Prime.mover/Sandbox
Welcome to prime.mover's Sandbox!
This space exists for prime.mover's storage and experimentation. Warning: This sandbox is not permanent. EXPECT ANYTHING YOU PUT HERE TO BE DELETED! |
This page exists for me to be able to test out features I am developing.
Contents |
Proof
test content. test content. test content. test content.
- $\displaystyle \int \frac 1 {1 + x^2} \mathrm{d}x = \arctan x + c$
All normal PW stuff works here, not just integrals ;)
Another proof title
You could even define another section inside (though I would discourage that)
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...Or was it?
The proofwiki editor prime.mover has begun this article and plans to complete it in due course.
I can't be arsed.
Please leave this page to be completed by prime.mover. It can be found in prime.mover's "In Progress" stubs.
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If radical deletion of existing material is planned, then please comment out the material to be deleted by enclosing it in comment delimiters <!-- -->. You are probably completely correct - but I'd like to compare my work with yours so as to learn what I did wrong.
The proofwiki editor prime.mover has begun this article and plans to complete it some time in the future.
Still to be done: Yeah yeah, I know, I haven't done it yet.
It can be found in prime.mover's Stubs.
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The proofwiki editor prime.mover has begun this article but has not completed it through lack of time, motivation or skill.
Problem: I'm a dumbo.
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It looks bare now, but wait till the furniture arrives.
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Why do I bother to get up in the morning?
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Please fix formatting and $\LaTeX$ errors and inconsistencies.
It may also need to be brought up to our standard house style.
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That Prime Mover geezer is well dodgy.
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That Prime Mover geezer is well dodgy.
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Try washing your socks occasionally.
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Did you get your hair cut like that for a bet?
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Template pages: stub, tidy, proofread, questionable, explain
$\newcommand{\Re}{\operatorname {Re}\,} \newcommand {\pFq} [5] {{}_{#1} \operatorname{F}_{#2} \left({\genfrac{}{}{0pt}{}{#3}{#4} \bigg|{#5} }\right)}$
We consider, for various values of $s$, the $n$-dimensional integral\begin{align} \tag{1} 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 (1) 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} \tag{2} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.\end{align}Appropriately defined, (2) also holds for negative odd integers. The reason for (2) was long a mystery, but it will be explained at the end of the paper.
The Elements (c. 300 B.C.E)
Definitions specific to this category can be found in Definitions/Abstract Algebra.
