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.
Herr Doktor Professor gg at Skulls in the Stars has been turning out a great deal of good physics blogging.
the writers over at Arcsecond posts physics problems occasionally too.
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.
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.
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.
This is a must read for all physicists! 🙂
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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 😉
I think it’s borderline blasphemy to not mention Riemann-zeta function before any other function..
http://en.wikipedia.org/wiki/Riemann_zeta_function
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-
Gavin: that is what as known as the “bump function” ; it is used in differential topology to “sew” things together.