úÎGËDª      NoneIConvergent when Re(z) > 0. The first argument is the c_n series to use  (G is an ineffecient but generic definition of the full infinite series. # Some precomputed finite prefix of & should be fed to this function, the 8 length of which will determine the accuracy achieved.) 4The c_n series in the convergent version of Stirling's approximation given  on wikipedia at  http://en.wikipedia.org/wiki/Stirling%27s_approximation#-A_convergent_version_of_Stirling.27s_formula  as fetched on 11 June 2010. 1The (signed) Stirling numbers of the first kind. 3The (unsigned) Stirling numbers of the first kind. HCompute the number of terms required to achieve a given precision for a K given value of z. The mamimum will typically (always?) be around 1, and L seems to be more or less independent of the precision desired (though not H of the machine epsilon - essentially, near zero I think this method is " extremely numerically unstable).  ! ! Safe-InferredCompute Lanczos': approximation to the gamma function, using the specified ) constants. Valid for Re(x) > 0.5. Use  or  to extend 8 to the whole real line or complex plane, respectively. Compute Lanczos'5 approximation to the natural logarithm of the gamma G function, using the specified constants. Valid for Re(x) > 0.5. Use    or  - to extend to the whole real line or complex  plane, respectively. IExtend an approximation of the gamma function from the domain x > 0.5 to  the whole real line. JExtend an approximation of the gamma function from the domain Re(x) > 0.5  to the whole complex plane. HExtend an approximation of the natural logarithm of the gamma function 1 from the domain x > 0.5 to the whole real line. HExtend an approximation of the natural logarithm of the gamma function 9 from the domain Re(x) > 0.5 to the whole complex plane. "#   "# None Factorial function  $%&'(    $%&'(None JContinued fraction representation of the lower incomplete gamma function. 6Lower incomplete gamma function, computed using Kummer' s confluent J hypergeometric function M(a;b;x). Specifically, this uses the identity: gamma(s,x) = x**s * exp (-x) / s * M(1; 1+s; x) From Abramowitz & Stegun (6.5.12). Recommended for use when x < s+1 INatural logarithm of lower gamma function, based on the same identity as  : and evaluated carefully to avoid overflow and underflow.  Recommended for use when x < s+1 VContinued fraction representation of the regularized lower incomplete gamma function. BRegularized lower incomplete gamma function, computed using Kummer's ; confluent hypergeometric function. Uses same identity as . Recommended for use when x < s+1 VContinued fraction representation of the regularized upper incomplete gamma function. # Recommended for use when x >= s+1 JContinued fraction representation of the upper incomplete gamma function. # Recommended for use when x >= s+1 DNatural logarithms of the convergents of the upper gamma function, 6 evaluated carefully to avoid overflow and underflow. # Recommended for use when x >= s+1 )Special case of Kummer'*s confluent hypergeometric function, used  in lower gamma functions. m_1_sp1 s z = M(1;s+1;z) *+,-)./012*+,-)./012None -Gamma function. Minimal definition is ether  or . ?The gamma function: gamma z == integral from 0 to infinity of  t -> t**(z-1) * exp (negate t) "Natural log of the gamma function &Natural log of the factorial function Incomplete gamma functions. ALower gamma function: lowerGamma s x == integral from 0 to x of  t -> t**(s-1) * exp (negate t) $Natural log of lower gamma function <Regularized lower incomplete gamma function: lowerGamma s x / gamma s HUpper gamma function: lowerGamma s x == integral from x to infinity of  t -> t**(s-1) * exp (negate t) $Natural log of upper gamma function <Regularized upper incomplete gamma function: upperGamma s x / gamma s 3BI have not yet come up with a good strategy for evaluating these  functions for negative x,. They can be rather numerically unstable. 4(This instance uses the Double instance.  5678349:;<    5678349:;<=      !"#$%&'()*+,-./0123456789:;<=>?@ABC gamma-0.9.0.2Math.Gamma.StirlingMath.Gamma.LanczosMath.Factorial Math.GammaMath.Gamma.IncompletelnGammaStirlingcssabs_sterms gammaLanczoslnGammaLanczosreflectreflectC reflectLn reflectLnC Factorial factorialGammagammalnGamma lnFactorial lowerGammaCFlowerGammaHypGeomlnLowerGammaHypGeompCFpHypGeomqCF upperGammaCFlnUpperGammaConvergentsIncGamma lowerGamma lnLowerGammap upperGamma lnUpperGammaq risingPowerscafractionalPart$fFactorialComplex$fFactorialDouble$fFactorialComplex0$fFactorialFloat$fFactorialIntegerm_1_sp1evalSignsignLog addSignLog negateSignLog log_m_1_sp1log_m_1_sp1_convergents interleavepow_x_s_div_gamma_s_div_exp_xpow_x_s_div_exp_x$fIncGammaDouble$fIncGammaFloatfloatGammaInfCutoffdoubleGammaInfCutoffcomplexDoubleToFloatcomplexFloatToDouble$fGammaComplex$fGammaComplex0 $fGammaDouble $fGammaFloat