Brad DeLong's Semi-Daily Journal (2004)

Yes, I think Abramowitz and Stegan might now be obsolete thanks to Mathworld, and the sites it links to. Just the other day I got an answer that used the confluent hypergeometric function of the first kind. But I had to code my formula into Matlab, which doesn’t have confluent hypergeometric functions as part of its built-in library of special functions. Mathworld to the rescue. With a few clicks I found: 0F1(;b;c) = Gamma[b] (-z)^(1-b)/2 J(b-1,2sqrt(-z). I could express my answer in terms of a Bessel function, which is in the Matlab built-in library. I was able to do this faster than looking it up in A&S.

BTW the study and teaching of special functions seems to be dying out. A lot of the new graduates have little or no knowledge of the whole area. They depend on packages, if it isn’t in the package they can’t do it.

There is another site which is very much worth a look, and which I prefer to mathworld. It is PlanetMath, at www.planetmath.org.

"BTW the study and teaching of special functions seems to be dying out."

Is this because numerical methods have replaced analytical approaches to differential equations? Or is it because all the useful ones have been found and are known?

Hey, dig this: Child Jesse Taylor has a new job... with Jerry Springer! The laughs just never stop.

“Is this because numerical methods have replaced analytical approaches to differential equations?”

I think that has a lot to do with it. Special functions are one way of cataloging solutions to differential equations, and now we can solve them numerically every time. And for the same reason, we don’t see as many asymptotic expansions. But with the ascendance of symbolic computing, we might see a turn around in the use of special functions. See the book “A=B,” Petkovsek et al which has to do with automatic computer proofs of hypergeometric identities. These things come up a lot in various probability calculations involving sums of binomials.

You're just now discovering that site? Not incidentally, I'm cited in one of the entries there. Which, in spite of the fact that I'm not thrilled with Wolfram's recent book, did please me, I confess.

Hi.
My name is Micah Wright. I'm a former Army Ranger, and I've been lying to you. I've kept the secret for years now, but all lies grow and eventually get out of control. This is me coming clean about my Big Lie. What did I lie about? Oh, nothing much...
Except that I was never an Army Ranger. I never served a day in a Ranger Regiment. I never went to Ranger School. The closest I ever got was Army ROTC.

Hi.
My name is Micah Wright. I'm a former Army Ranger, and I've been lying to you. I've kept the secret for years now, but all lies grow and eventually get out of control. This is me coming clean about my Big Lie. What did I lie about? Oh, nothing much...
Except that I was never an Army Ranger. I never served a day in a Ranger Regiment. I never went to Ranger School. The closest I ever got was Army ROTC.

Hi.
My name is Micah Wright. I'm a former Army Ranger, and I've been lying to you. I've kept the secret for years now, but all lies grow and eventually get out of control. This is me coming clean about my Big Lie. What did I lie about? Oh, nothing much...
Except that I was never an Army Ranger. I never served a day in a Ranger Regiment. I never went to Ranger School. The closest I ever got was Army ROTC.

I love that site, especially the section on map projections.

Another good math site focused on preparing students for math competitions is:

http://www.artofproblemsolving.com

They offer Internet classes that bring top math students from around the world together.

I don't think the study of special functions is dying out. It's no longer pursued at the same level that it once was (probably because of numerical solutions, as some have already speculated above), but the amount of knowledge is still growing. For example, there are new classes of special functions, such as q-hypergeometric functions. People have been working on fitting special functions into larger frameworks -- for example, many of the classical special functions occur as matrix entries in matrix Lie groups.

Walt Pohl: That’s great news. Do you have a reference on q-hypergeometrics?

Try wikipedia. It doesn't contribute to the illusion that WOlfram knows anything about anything, except being an outstanding businessperson. I'll grant him that.

Wikipedia is not as comprehensive as Mathworld. Try looking for “hypergeometric” and you won’t get much more than the distribution and some discussion about series. No identities, nothing on confluent hypergeometric functions etc. You still need A&S or Mathworld (including it’s links). I share your misgivings about Wolfram, especially “A New Kind of Science.” But what product compares to Mathematica? Maple is ok, but not in the same league. Matlab has limited symbolic capability, and it’s really deficient in the special function area, and much less flexible as a language. The numerics in Mathematica is superior to everything else I know. On the other hand Wolfram Research is not good at admitting to mistakes, let alone fixing them quickly. Mathematica version 5 has some bugs not in version 4.2. But I still use version 5 because of the new features and capabilities.

"Wikipedia is not as comprehensive as Mathworld"

True. But you could easily help change that.

"But what product compares to Mathematica"

Depends of course what you want to do. In of course our friend *** ****** in the ********* dept isn't helping with his promotions of mathematica.

"A New Kind of Science."

I'm all for crackpot science, up to a point. In fact, universities, at least private ones should have a fellowship called something like:

"The Creationist and Crackpot fellowship".

If properly managed and endowed, it will keep Creationists and Crackpots more in the mainstream and keep everybody else on their toes. And who knows, the crackpots could be right,.. sometimes.

Wolfram, he doesn't need it but he should fund it for others.

There's a page on Mathworld on q-hypergeometric series:

http://mathworld.wolfram.com/q-HypergeometricFunction.html

The credit for Mathworld should go to Eric Weisstein, not Wolfram.

"The credit for Mathworld should go to Eric Weisstein, not Wolfram."

I agree. Hopefully, my remark was not interpreted as asserting the contrary.

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interesting discussion guys