Approximating the Gamma function via even-symmetric series without use of log or exp (but using sqrt and cosine) ==========================Warren D Smith===January 2012============================= Let F(x) = 1/GAMMA(x+1/2). [Half shift and reciprocation.] Then the even-symmetric product F(x) * F(-x) = cos(Pi*x)/Pi is known. If we know both the product and sum we can solve for the function value itself: a*b = p, a+b = s has solution: 1/a = 1 / [ s/2 +- sqrt(s*s/4 - p) ] = [ s/2 -+ sqrt(s*s/4 - p) ] / p Note for us the +- sign changes when x's sign changes, thus preventing F(x) from wrongly being even symmetric. So let us focus our efforts from here on on the even-symmetric sum-function S(x) = F(x) * F(-x). Note S(0) = 2/sqrt(Pi) = 1.128379..., S(1/2) = S(-1/2) = 1. Even-symmetric Maclaurin series: S(x) = a0 + a2*x^2 + a4*x^4 + ... n a[n] (probably to too many sig figs) 0 +1.1283791670955125738961589031215451716881012586580 2 -0.60900348841611068216370238307250126595327706534976 4 +0.40117091233557519976494358646090191877402434208538 6 -0.077697440910247074709804615656387720872711949327276 8 +0.0031307326305509859549042071031235561394059839823241 10 +0.00033007044645876790960938365344658406871455958329299 12 -0.000025637408145313339590477391674859646679049609798940 this series has radius of convergence infinity and gives you over 1 decimal per term, once it revs up. Not bad. 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 S(x) = b0 + b2*T2(2x) + b4*T4(2x) + b6*T6(2x) + ... n b[n] (probably to too many sig figs) 0 1.06128021416184195755744883511279486310105592266570662506060510783873 2 -0.64152437934192797229552498479759593618768485117359381613700427217805e-1 4 0.29092677526038089329719837630334817453102073653264336290827255795697e-2 6 -0.3714620647551524243440434068657199748975962838655258078708791617057e-4 8 0.10163636208388614086441283321477130413815337257574084383480844642e-6 10 0.59291158040245953231577074770537791489904680950435069737753226e-9 12 -0.305157867631736055686720568881723513798138400380975541810644e-11 14 0.44513545729495127881935456940665312188830441896508085535e-15 16 0.1524786364625270677102897188967959495046769658412825416e-16 18 -0.1400941201553971044962512532302471662071644116029698e-19 20 -0.2851745012755828389691716435359123195188528447288e-22 22 0.3600820470720857936809914097648872787060815612e-25 24 0.2639903369752602613572830229951948536575891e-28 26 -0.4098407309916913596589789736282126814407e-31 28 -0.1452740903699932514280624009694405059e-34 30 0.2640523100792403872575823849289582e-37 32 0.563463271407374814567635972279e-41 34 -0.1078349899717743292405719179e-43 36 -0.178398654119405519004237e-47 38 0.297602323431721756704e-50 40 0.48138810046683331e-54 Incidentally, TnStar(z^2) = Tn(2z^2-1) = T[2n](z).