I’ve not encountered a Dr Hate yet, but some lit searching on W-R stars uncovered a potentially terrifying member of one’s thesis committee, Dr Doom:

http://adsabs.harvard.edu/abs/1988A%26A…192..170D

P. Colwell, Bessel Functions and Kepler’s Equation, Amer. Math. Monthly, 99, 45 (1992), available on JSTOR as http://www.jstor.org/stable/2324547

However, Colwell does mention that “[Bessel] didn’t consider [Kepler's equation] the most important of the problems of celestial mechanics which lead [sic] him to [his functions],” so it’s unclear whether this is Bessel’s *original* derivation, though the letter to Olbers in which he announced the Bessel function solution to Kepler’s equation predates the paper by eight yeasrs.

But I certainly don’t share ObsessiveMathFreak’s distaste for hypergeometric functions, or even Maclaurin series representations in general–their overuse only points to their power in certain applications, so people are apt to apply them unthinkingly to other problems for which a somewhat different approach might be much more expedient (or at least slicker). This is particularly true of Maclaurin series, and is something I have found myself prone to (albeit without the Frobenius method).

However, the practice of just “regurgitating … Mathematica output” that Pieter Kok mentions is definitely to be deplored: Anyone doing this sort of work will almost surely get a better appreciation for the problem at hand if they at least attempt to reproduce Mathematica’s results with pencil-and-paper. And it’s criminal not to at least try to massage the output from Mathematica (or Maple, or even pencil-and-paper…) into a nicer form before putting it in print, though I fear that doing so is not always the common practice it should be.

Finally, I imagine that the usual practice of defining common special functions using differential equations is adopted to provide a relatively unified framework for introducing them, without necessarily having to give the more specialized historical details of the problems the functions were initially introduced to solve. But even then, this isn’t done blindly–I’ve never seen the gamma, theta, zeta, or error functions defined using differential equations, for instance.

The ultimate artefact of this process has been the pages of Wolfram’s Mathworld, and to a lesser extent Mathematica itself. The contemporary quest for meaning in hypergeometric functions is undoubtedly related to Mathematica’s propensity to trot them out for even the simplest of problems. I shudder to think of the thousands of man hours wasted deriving and working with these utterly intractable functions. I doubt this was the kind of work Abramowitz or Stegun ever intended to promote.

I’ll finish by saying that if anyone has a link to an English language version of Bessel’s original derivation of the functions that bear his name(derived in the context of solving Keplers equation; and which is probably very closely related to Anger functions), I would be eternally grateful, and I suspect that more than few readers might be interested to learn that there is life beyond the Handbook of Mathematical Functions.

Bessel’s original paper was:
Untersuchung des Thiels der planetarischen Storungen, welcher aus der Bewegung der Sonne entsteht, Abh. Akad. Wiss. berlin, math, Kl, 1824(1826)
But there’s probably a more modern exposition of the same somewhere (though probably not after A&S’s publication).

I received an email linking to the URL below from my sister this morning – I imagine it’s making the rounds, but I recall no mention of it here.

http://ig.unb.br/prof/ReinhardtAdolfoF**k.htm

Replace the asterisks exactly as you imagine they should be replaced – I didn’t want this to get caught in any spam filter.

http://en.wikipedia.org/wiki/Kelvin_functions

And don’t call him “Dick”, please.

And Strether — I have never really found an organizing principle to A&S, which is why I tend to prefer the book form, which is easier to flip though. My flipping process is either somewhat guided (“This looks something like a Bessel function. Let’s start there…”) or nearly random (“Is there anything on this page that has hyperbolic sines in it?”). Also, I’m Julianne, not Sean. Not that being Sean is a bad thing, and while I’d be happy to take his car, none of his clothes would fit.