Approximating the Gamma function via odd-symmetric series without use of log or exp ==========================Warren D Smith===January 2012============================= You can also make the same approach work but now odd-symmetrically. Let F(x) = 1/GAMMA(x+1/2) [Half shift and reciprocation] as before. Then the even-symmetric product F(x) * F(-x) = cos(Pi*x)/Pi is known. If we know both the product and difference we can solve for the function value itself: a*b = p, a-b = d has solution: 1/a = 1/ [ -d/2 +- sqrt(d*d/4+p) ] = p / [ d/2 +- sqrt(d*d/4+p) ] So let us focus our efforts from here on on the odd-symmetric difference-function d(x) = F(x) - F(-x). Note d(0) = 0 , d(1/2) = -d(-1/2) = 1. Odd-symmetric Maclaurin series: d(x) = a1*x + a3*x^3 + a5*x^5 + ... n a[n] (probably too many sig figs) 1 +2.2155838077457420463420706011273861532011964034196 3 -0.8782068450071543970312933271566146210547472756753 5 +0.0597785511268765525562112629760526583628122029271 7 +0.0153465270962211928891443648520834538796743000868 9 -0.0020691028884223045620734315390504392488606408315 11 +0.0000368136041297145745345391282448961611113354311 this series has radius of convergence infinity. The coefficients are expressible in closed form, MAPLE knows how in terms of gamma, Pi, Riemann zeta, ln... but the formulas seem pretty nasty. Chebyshev series valid for x real with |x|<1/2 d(x) = b1*T1(2x) + b3*T3(2x) + b5*T5(2x) + b7*T7(2x) + ... n b[n] (probably too many sig figs) 1 1.026691148712964028143010336735095933411651823011179165757743251080660 3 -0.268221685958169797573564305029379545327023334950301175003624523012322e-1 5 0.129303162744591326555007007441325773066547121152841795798802532378070e-3 7 0.1732283652500262119684594728924325257327912284564252029346753804011e-5 9 -0.15582885160481449158005134567946714001039426528884456840721825850e-7 11 0.19215614127272524033878302481695622873236838653216308436905995e-10 13 0.125710846652610707719760530698850454093189146137737892751004e-12 15 -0.304239287982118987546845675936710370189616450492341901499e-15 17 -0.228543304071050074008778140993527912584044876007789732e-18 19 0.1026747421737886955434375472401652907750438705373650e-20 21 -0.3428129092655546076050020932259010193356887992e-25 23 -0.1521613621112612210432980081809452529820278380e-26 25 0.41130441677834000795641588393066361560497e-30 27 0.125353496337946321379878889536091985234e-32 29 -0.426948536716900856438110447660012316e-36 31 -0.654564249588552321735272510074462e-39 33 0.21758765315458092372227254347e-42 35 0.234927802069331038492894961e-45 37 -0.6580132252204947636736e-49 39 -0.6078105895687311484e-52