On the subject of the gamma function, I'd be interested to know if I got this correct for complex gamma - it's the func I wrote for the Ultra Fractal math library = obviously it's not the most accurate implementation but I was balancing speed with accuracy and in adition it was taking too long to hunt down a better table of values to use ;) *************** ; Complex Gamma() David Makin June 2008 ; @param pz complex argument ; @return Gamma function of argument static complex func ComplexGamma(complex pz) complex w = pz if pz==0.0 w = recip(0) else if real(pz)<0.0 w = -pz if imag(pz)==0.0 if (real(pz)%1)==0.0 w = recip(0) endif endif endif if real(w)<1e300 w = (sqrt(2.0*#pi)* \ (0.99999999999999709182 \ + 57.156235665862923517/(w+1.0) \ - 59.597960355475491248/(w+2.0) \ + 14.136097974741747174/(w+3.0) \ - 0.49191381609762019978/(w+4.0) \ + .33994649984811888699E-4/(w+5.0) \ + .46523628927048575665E-4/(w+6.0) \ - .98374475304879564677e-4/(w+7.0) \ + .15808870322491248884E-3/(w+8.0) \ - .21026444172410488319e-3/(w+9.0) \ + .21743961811521264320e-3/(w+10.0) \ - .16431810653676389022e-3/(w+11.0) \ + .84418223983852743293e-4/(w+12.0) \ - .26190838401581408670e-4/(w+13.0) \ + .36899182659531622704e-5/(w+14.0)) \ /w) * exp(-w-5.2421875)*(w+5.2421875)^(w+0.5) if real(pz)<0.0 w = pz*w*sin(#pi*pz) if (w==0.0) w = recip(0) else w = -#pi/w endif endif endif endif return w endfunc ************** On 28 Dec 2011, at 18:13, Simon Plouffe wrote:
Hello,
the formula for (1/4)! is quite interesting, the approximation is 88 digits, this is unique.
Now about Gamma(n/48), I do not think that any higher value would lead to simple approximations as mr Gosper showed, as the index increases : it gets quite messy, we can only hope to get the first denominators in my opinion.
Nevertheless, the values for (1/3)! and (1/4)! are impressive, the technique uses <approximations> of dedekind functions, a neat trick.
best regards, Simon Plouffe
Le 28/12/2011 18:43, Joerg Arndt a écrit :
* Warren Smith<warren.wds@gmail.com> [Dec 28. 2011 18:30]:
[...]
Then I tried Gosper's (1/4)! and I don't understand how Gosper got his formula. Actually, I practically never understand how Gosper gets his formulas. As I mentioned before, it'd be nice to find formulas for GAMMA(k/48) for example. Such formulas might exist in terms of algebraic numbers and the AGM.
Just as quick copy and paste (suggest to start with the last one):
J.\ M.\ Borwein, I.\ J.\ Zucker: {Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind}, IMA Journal of Numerical Analysis, vol.12, no.4, pp.519-526, \bdate{1992}.}
Greg Martin: {A product of Gamma function values at fractions with the same denominator}, arXiv:0907.4384v1 [math.CA], \bdate{24-July-2009}. URL: \url{http://arxiv.org/abs/0907.4384}.}
Albert Nijenhuis: {Small Gamma Products with Simple Values}, arXiv:0907.1689v1 [math.CA], \bdate{9-July-2009}. URL: \url{http://arxiv.org/abs/0907.1689}.}
Raimundas Vid\={u}nas: {Expressions for values of the gamma function}, arXiv:math.CA/0403510, \bdate{30-March-2004}. URL: \url{http://arxiv.org/abs/math/0403510}.}
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