From Littlewood's Miscellany: A minute I wrote (about 1917) for the Ballistic Office ended with the sentence `Thus σ should be made as small as possible'. This did not appear in the printed minute. But P. J. Grigg said, `What is that?' A speck in a blank space at the end proved to be the tiniest σ I have ever seen (the printers must have scoured London for it).
Modern editions of Gradshteyn and Ryzhik, 8.338 (5.) give the utterly miscellaneous Product[Gamma[k/3], {k, 8}] == 640/3^6 (\[Pi]/Sqrt[3])^3 (Ascribed to WH. An exercise in Whittaker and Watson?) The first English edition absurdly has Product[Gamma[k/3], {k,∞)] presumably photocopied from the Russian original. The next English edition (which I can find neither online nor in my basement) has the formula obviously corrected by scissoring out the ∞ and turning it sideways. (Apropos my recent rantings, G&R has some exotic "ensemble sums" ascribed to JO (Jolley's Summation of Series?) which are completely bogus. More on this later, hopefully.) Watson's Treatise on the Theory of Bessel Functions, 17·6(3) has Sum(n^2*bessel_j[n](n*z)^2,n,1,inf)=z^2*(4+z^2)*(1-z^2)^(-1/2)/16 Empirically, it should be Sum(n^2*bessel_j[n](n*z)^2,n,1,inf)=z^2*(4+z^2)*(1-z^2)^(-7/2)/16 Presumably Watson had it right, but the printer found the exponent 7/2 too improbable. Additional clue: 17·6(2) also shows a factor of (1-z^2)^(-1/2), but displayed with a radical instead of parens. (I swear I once saw √(1-x²) called the radiculus function. Seems worth resurrecting.) (c19) SUM(N^2*BESSEL_J[N](N*Z)^2,N,1,INF) = Z^2*(4+Z^2)*(1-Z^2)^-(7/2)/16; Time= 0 msec. inf ==== 2 2 \ 2 2 z (z + 4) (d19) > n bessel_j (n z) = -------------- / n 2 7/2 ==== 16 (1 - z ) n = 1 (c20) TAYLOR(%,Z,0,11); Time= 140 msec. 2 4 6 8 10 z 15 z 35 z 525 z 3465 z (d20)/T/ -- + ----- + ----- + ------ + -------- + . . . = 4 16 16 128 512 2 4 6 8 10 z 15 z 35 z 525 z 3465 z -- + ----- + ----- + ------ + -------- + . . . 4 16 16 128 512 (c21) BLOCK([FANCY_DISPLAY : FALSE],PLAYBACK([19,21],TIME)); (Note Macsyma here. After all these decades, Mathematica's Series is still braindead.) --rwg