I sometimes forget that we don’t all read the same blogs, and that it’s good to recommend some of the fun stuff out there on the internets. So let me give a shout-out to Matt Springer at Built on Facts, who had the brilliant idea of discussing a different function every Sunday. Functions are one of those things that are as necessary to math and science as breathing, but which don’t necessarily percolate into the wider world. And he (quite correctly, I think) interprets his self-imposed mandate fairly liberally, taking the time to talk about various issues in middle-level mathematics. Here are some selections from Matt’s series:
- Exponential
- Arctangent
- Witch of Agnesi
- Stirling’s approximation
- Continuous but almost-nowhere differentiable
Consider this an open thread to recommend other stuff we should all be reading. Or your favorite functions.
March 1st, 2009 at 11:20 am
Herr Doktor Professor gg at Skulls in the Stars has been turning out a great deal of good physics blogging.
March 1st, 2009 at 1:11 pm
the writers over at Arcsecond posts physics problems occasionally too.
March 1st, 2009 at 4:31 pm
The cusp geometry is a weird function that loses stability and jumps between two solution sets. It’s used in economics to model transitions from monopolism to competition in a market.
March 1st, 2009 at 4:48 pm
Here is mine:
g(x) = exp(-1/x^2) for x not equal to zero
= 0 if x = 0
g is “C-infinity” in that it has derivatives of all orders everywhere but is not analytic at x = 0, which means it doesn’t have a Taylor series about x = 0 that is valid on any open interval containing zero.
This is the basic building block of the “bump function” that allows us to do “surgery” in differential topology but the stumbling block between extending results from smooth to analytic manifolds.
March 1st, 2009 at 7:34 pm
Glad you like it! I have to say it’s one of my favorite things to write – there’s no limit to the material, and it’s all fascinating.
March 2nd, 2009 at 7:46 am
This is a must read for all physicists!
March 2nd, 2009 at 8:07 am
[...] Cosmic variance has a thread on people’s favorite mathematical functions. Here is one of mine: for [...]
March 2nd, 2009 at 1:48 pm
And then there’s Ackermann’s function, which I covered on february 10th, and the devil’s staircase which I’ll cover sometime this month. The latter has a zero derivative almost everywhere, except at the cantor dust points where it is infinite. It climbs from 0 to 1 ONLY at the cantor points. Continuous but not differentiable
March 2nd, 2009 at 2:13 pm
I think it’s borderline blasphemy to not mention Riemann-zeta function before any other function..
http://en.wikipedia.org/wiki/Riemann_zeta_function
March 3rd, 2009 at 2:28 am
My favourite is (does it have a name?):
f(x) = 1 /(1 – e(-1/x))
lim f(x) = 1
x -> 0+
lim f(x) = 0
x -> 0-
March 3rd, 2009 at 4:59 am
Gavin: that is what as known as the “bump function” ; it is used in differential topology to “sew” things together.