Two nice integrals for derangements

You can find the full post at my numbersandshapes.net blog, but if you don’t want to follow the link, the integrals (for D_n, the number of derangements of order n, are:

\displaystyle{D_n=\int^\infty_0 e^{-x}(x-1)^n\,dx}

and

\displaystyle{D_n=\int^1_0(-1-\ln(x))^n\,dx}

This last integral can be equilavently written, of course as

\displaystyle{D_n=(-1)^n\int^1_0(1+\ln(x))^n\,dx}

or

\displaystyle{D_n=\left\vert\int^1_0(1+\ln(x))^n\,dx\right\vert}

I think that derangements having an integral representation is rather splendid!

Neusis constructions (3): the regular heptagon

Constructing a regular heptagon is a very elegant and delightful exercise in geometry and algebra, and which requires the trisection of an angle.  There are plenty of discussions all over the web but mine, which is available at Numbers and Shapes, I think is the clearest, with its LaTeXed mathematics, and diagrams drawn with TiKZ.

Neusis constructions (2): trisections

This post, containing a nice slew of methods of trisecting a general angle using neusis methods, can be found at Numbers and Shapes.

Neusis constructions (1)

You can find this post, introducing geometric constructions which allow the use of a straight-edge with two marks, at my new site Numbers and Shapes.

Solving a cubic by folding

This post, which is on my new site here:

http://www.numbersandshapes.net/?p=2520

shows how a cubic equation can be solved by origami.  This is not a new result by any means, but it’s hard to find a simple proof of how the construction works.

An alternative to partial fractions

The full post is available at my new site:

http://www.numbersandshapes.net/?p=2495

Enjoy!

The foolishness of signature images

As I explained on the page Numbers and Shapes, this blog is moving to a new home, and you’ll find this post there: The foolishness of signature images

Enjoy!

The AIM Network

The Australian Independent Media Network

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