Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: community: general

Topic: Ramanujan

John Baez (Sep 03 2020 at 22:31):

I enjoyed Robert Kanigel's biography of Ramanujan, but I decided to really understand Ramanujan better I should try to understand one of the formulas he sent to Hardy in his first letter.

I picked what seemed to be the easiest one, and I got led on an interesting chase:

• Chasing the Tail of the Gaussian (Part 1).

• Chasing the Tail of the Gaussian (Part 2).

It starts with a puzzle Ramanujan posed in 1914:

Prove that

$\left(\frac{1}{1} + \frac{1}{1 \cdot 3} + \frac{1}{1 \cdot 3 \cdot 5} + \cdots\right) \; + \; \frac{1}{1 + \frac{1}{1 + \frac{2}{1 + \frac{3}{1 + \frac{4}{1 + \frac{5}{1 + \ddots}}}}}} = \sqrt{\frac{\pi e}{2}}$

Nikolaj Kuntner (Sep 04 2020 at 16:35):

There's also continued fractions for $\exp$, so is there also one for just $\int_x^\infty \exp(-\tfrac{1}{2}t^2){\mathrm d}t$?

Also, I've heard bad mouthing that many of the funky Ramanujan series formulas are just results of series speedup principles being applied to nice formulas.

Nikolaj Kuntner (Sep 04 2020 at 16:37):

As in https://en.wikipedia.org/wiki/Series_acceleration

John Baez (Sep 04 2020 at 21:42):

Nikolaj Kuntner said:

There's also continued fractions for $\exp$, so is there also one for just $\int_x^\infty \exp(-\tfrac{1}{2}t^2){\mathrm d}t$?

I don't know one, but that doesn't mean much. There has been a lot of work on approximating this integral for large $x$, but since this integral it's asymptotic to $\frac{1}{x} \exp(-\tfrac{1}{2}x^2)$ as $x \to +\infty$, this work naturally focuses on approximating the function

$\phi(x) = \exp(\tfrac{1}{2}x^2) \int_x^\infty \exp(-\tfrac{1}{2}t^2){\mathrm d}t$

You can read an overview here.

Nikolaj Kuntner (Sep 04 2020 at 22:47):

k, thx.
(one minus sign too much there in the first exponent)

Nikolaj Kuntner (Sep 04 2020 at 22:48):

Maybe you'll find a continued fraction representation :)
I think at one point I justified to myself why continued fraction are interesting, but generally they never seem to pop up in physics.

John Baez (Sep 04 2020 at 23:12):

I fixed that exponent.

John Baez (Sep 04 2020 at 23:13):

Continued fractions do show up in physics, e.g.

• Wikipedia, Method of continued fractions.

John Baez (Sep 04 2020 at 23:15):

I think mainly not enough physicists know them well enough to use them. They have some advantages, e.g. the radius of convergence of $\ln(1+x)$ is just $1$, but the corresponding continued fraction works everywhere on the complex plane except the negative real axis. I think I've seen "Pade approximants", which are related to continued fractions, used in quantum field theory to get around some problems with diverging power series.

John Baez (Sep 04 2020 at 23:16):

But I'm just learning how to use them!

John Baez (Sep 06 2020 at 06:35):

I wrote a little intro to different stages in the history of continued fractions:

• Three phases of continued fraction theory.

The ancient Greeks, Euler and Gauss!

Amar Hadzihasanovic (Sep 06 2020 at 10:31):

@John Baez “Dusko Pavlovic” got compressed into “Duskovic” in your post :)

John Baez (Sep 06 2020 at 16:30):

Weird!