The road to L

For my tentative first post here I'm going to be lazy and recycle the first post from my old blog, just to see how it comes out.

(Originally posted 090502, last revised 090513)

It was with good intentions, or at least neutral intentions, that I started down the road that was to take me to L. I certainly wasn’t looking for L ; in fact, I had no idea it existed until I found it -- or rather, I should say, one of them, because there are in fact many Ls.

Here, approximated (?) to 50 digits, is the first L I found, which I rashly concluded was the only one:

0.3181315052047641353126542515876645172035176138714 +1.3372357014306894089011621431937106125395021384605i

The location of this number on the complex plane is shown roughly in the graphic above, with scale indicated by the unit circle in white.

I stumbled on it when I started evaluating log log log . . . log z for various complex numbers z. Quite to my surprise, iterating the log operation invariably led to this number, regardless of the value of z. (I started out evaluating iterations of ln with the Google Calculator, which revealed the unexpected convergence, and then I switched to Casio’s Keisan site, which allowed me to determine the value more precisely.)

I figured this must be a Very Special Number Indeed, and I decided to dub it “ L,” for “logarithmic limit. However, two mathematicians I consulted breathlessly concerning this Great Discovery patiently informed me that that it is just one of an apparently infinite set of such numbers, all satisfying the condition z = e^z. So it is far from unique. But I still think that it and its fellows are interesting and, yes, very special. I shall write more about them another day.