Monday, May 17, 2010

Abramowitz/Stegun goes online

Did you ever need to learn about the properties of some obscure mathematical function which turns up when you try to solve, say, the Schrödinger equation with a linear potential?

In the times before Wikipedia and Eric Weisstein's World of Mathematics/MathWorld, the usual way to proceed was to go to the library and look up in the "Abramowitz/Stegun", a compilation of formulas, relations, graphs and data tables for all kinds of functions you can think of.

Airy functions Ai(x), Bi(x) and M(x).

Over the last years, Milton Abramowitz' and Irene A. Stegun's time-honored "Handbook of Mathematical Functions" has been carried over to the internet age as the Digital Library of Mathematical Functions. Published by the US National Institute for Standards and Technology (NIST),

... the NIST Digital Library of Mathematical Functions (DLMF), is the culmination of a project that was conceived in 1996 at the National Institute of Standards and Technology (NIST). The project had two equally important goals: to develop an authoritative replacement for the highly successful Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, published in 1964 by the National Bureau of Standards (M. Abramowitz and I. A. Stegun, editors); and to disseminate essentially the same information from a public Web site operated by NIST. (From the DLMF Preface)

Parts of the DLMF have been available since some time, but the complete site went online just last week, on May 11.

In comparison to the old printed book, there are more functions and formulas, which all can be copied as Latex or MathML code. And while the function graphs at MathWorld are interactive, the DLMF features more detailed descriptions of applications in mathematics and physics, and links to freely available software libraries.

Should I ever need to code Jacobian Elliptic Functions, I'll know where to look them up.

Via bit-player, where you can also read more about the history of the Abramowitz/Stegun.


  1. Or, you could search WolframAlpha. You can find the Jacobi elliptic equations here:

    Clicking any one takes you to a page where you can read about it and plot it.

  2. @Bryan
    I think you meant:

    The links then take you through to either the mathematica documentation, the functions site or mathworld. The functions site is probably the most useful and contains a heap of identities and relations for the functions. It's been my online replacement for A/S for the last few years.

  3. Thanks for letting us know!

    Question: I've tried at some point to find online integral tables, but haven't found much. Does anybody know a good resource? I know Maple/Mathematica can do a lot of integrals even analytically, but the result isn't always useful and sometimes I need the fine print. Best,


  4. Hi Stefan,

    great news!

    Best, Kay

  5. Igor: Was that a reply to my question? Dunno, I've turned Wolfram upside down (granted, that was 2 years back or so), but really couldn't find things like integral over products of incomplete gamma functions or exp with a function in the argument etc. I mean, I have two books with integral tables (yes, it's some Russians, but I forget the names, it's the standard tables you find in any library), but I don't usually take them with me when I travel, thus my question. I've found parts of them online (needless to say that were the parts I didn't need), so I've been wondering since if there isn't a more complete site with integral tables. Best,


  6. Hi Bee,
    The famous book: 'Table of Integrals, Series, and Products' by Gradshteyn and Ryzhik can be bought in CD format. You could take it with you or copy it on your notebook.


  7. Right, Gradshteyn, that's the book that I have (two volumes actually)! So I'll have to buy a CD... pooh. And my laptop doesn't have a CDrom, so what's the point?

  8. Bee,

    Find a computer with a CD drive and rip it to a flash drive? That'd work I'd imagine.

  9. Sure, but it's so cumbersome. Why not just have a website online, if necessary one that charges a fee?

  10. WolframAlpha relies, as far as functions and their properties and relations are concerned, on MathWorld and MathWorld, at least, usually refers to the Abramowitz/Stegun, and my impression is that the DLMF is more comprehensive than the Wolfram collection, albeit not as fancy, and without the impressive interactive plotting functionality of the Wolfram sites...

    On the other hand, it is easy to copy Latex code of formulas from DLMF, and I would have appreciated the links to the function libraries in Fortran and C that they now offer when I had to do a few numerical calculations involving Bessel and Airy functions.

    BTW, has indeed a few integrals, e.g. here for the Incomplete Gamma Function. No idea how this compares with the classical tables such as Ryzhik Gradshteyn.

    I have just seen, the latest edition of the Ryzhik Gradshteyn is searchable via amazons "look inside".

    BTW, does someone have some experience with ?

    Cheers, Stefan

  11. Bee,

    No idea. I'd imagine the demand is simply not that great. Out of curiosity, what parts of the tables do you need? As in, why doesn't Maple/Mathematica not sate your need?

  12. Stefan,

    I've used Wolfram Integrator. It works quite nicely for what I've needed to do. I haven't tried anything too complicated however but it definitely is able to handle most integrals I plug into it.

  13. Flash drives have an unexpected benefit: Homeland Severity seize and copy them. You can vote 16 GB of "no" every time you pass through a US airport. A necklace of high capacity flash drives brimming with gibberish is the right thing to do.

    Intelligence requires constant preening, but stupidity is an engine of its own creation. Gorge the congenitally inconsequential.

  14. Thanks Stefan,

    I added it to my Favs. (Now I have 4085 Favs in 50 root folders and 667 sub-folders, lol.)

    Many year ago, I created an equation for a point in 3D, F(x,y,z), such that it would be zero everywhere in space, except for a specified (x1, y1, z1) where F(x1,y1,z1) would have the value of 1. I used all natural, HS-level math functions to do it. I wonder if there's an equation for a point is at the NIST site. I'll look sometime.

    What was curious, is that I could make the volume of the point arbitrarily small, but never could get it to be exactly zero. That made me wonder if a zero-dimensional, zero-volume point had any meaning or reality. I concluded that in reality, any "point" had to be a "fuzzy point" with arbitrarily small but non-zero size, since that was the limitation I could not overcome mathematically.

    Also, it was easy to add a time dimension and have the point spin in any one, or two or three orthogonal directions; although that seemed odd to have a point spin.

    Oh, there was a problem: the point (magnitude +1) always came with an anti-point (magnitude -1), and six other quasi-points ("ghost points" as I called them, since they had a markedly different form and not fully 3-D structure to them ... curiously, the magnitudes of the six quasi-points were: +1/3, +1/3, +2/3 and -1/3, -1/3, -2/3). The extraneous points were annoying to me, since I just wanted the equation to have a value (magnitude) of 1.0 at only one specific point in space. But I couldn't eliminate the anti-point and 6 quasi-points from the equation ... they were an inevitable part of equation. So I modified the equation such that the anti-point and 6 quasi- (ghost) points were always at an infinite distance from the point at (x1, y1, z1) ... so that then, in a sense, they did not exist.

    Still, when I had the point spin, I couldn't help but realize that its anti-point and six quasi-points were, by necessity, also spinning, in infinite circles, out at an infinite distance away. Not too elegant. lol. But it was the best I could do.

    About 15 years later, Mathematica became available, so I was able to enter the equation into the program and plot it out. It plotted out as an arbitrarily small point with magnitude 1.0, just as it was designed to, which was gratifying to see.


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  17. Hi Stefan,

    Thanks once again for being as you so often are the heralder of good news. However, self admittedly this won’t be something I would find reason to use much, yet it certainly expands further what is available to anyone having access to the web. It has me to wonder if there ever will come a day where a physics researcher will be hired where in the section of their CV referring to education it will simply read “WWW”.? It sounds unlikely I know, yet perhaps this has more to do with our antiquated concepts of education catching up with our newly expanded potentials more than anything else.



  18. Hi Luke,

    Maple/Mathematica will sometimes just tell you an integral doesn't converge or simply spit out the same integral as result, neither of which is helpful. I'd sometimes actually need to know for which cases it does when converge and rather than trusting Maple that it doesn't know an integral I'd rather look it up myself (doesn't happen too often though). Best,



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