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|newpage| Gamma functions =============================================================================== Sign of the gamma function ------------------------------------------------------------------------------- .. method:: math53.signgamma(x) Returns the sign of `\Gamma(x)`, which is `+1` if `x > 0` or if `\lfloor x \rfloor` is even, `-1` otherwise, and meaningless for `0` or negative integers. See also Wikipedia :cite:p:`WikipediaFun75`, MathWorld :cite:p:`WolframFun75`, NIST :cite:p:`DLMFun75`, BoostMath :cite:p:`BoostFun75`, :cite:t:`Ehrhardt2018` (3.5.1.9). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.SignGamma(1.5) ereal('5.2359877559829887307E-1') >>> ereal.SignGamma('1.51') ereal('5.3518479027559984754E-1') Logarithm and sign of the gamma function ------------------------------------------------------------------------------- .. method:: math53.lgamma_s(x,s) Returns (as a tuple) the logarithm and sign of the gamma function. See also Wikipedia :cite:p:`WikipediaFun75`, MathWorld :cite:p:`WolframFun75`, NIST :cite:p:`DLMFun75`, BoostMath :cite:p:`BoostFun75`, :cite:t:`Ehrhardt2018` (3.5.1.10). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.LogGammaS(1.5) ereal('5.2359877559829887307E-1') >>> ereal.LogGammaS('1.51') ereal('5.3518479027559984754E-1') Logarithm of `\Gamma(1 + x)` ------------------------------------------------------------------------------- .. method:: math53.lgamma1p(x) Returns `\log|\Gamma(1+x)|` with increased accuracy for `x` near `0`. See also Wikipedia :cite:p:`WikipediaFun77`, MathWorld :cite:p:`WolframFun77`, BoostMath :cite:p:`BoostFun77`, :cite:t:`Ehrhardt2018` (3.5.1.7). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.LogGamma1p(1.5) ereal('5.2359877559829887307E-1') >>> ereal.LogGamma1p('1.51') ereal('5.3518479027559984754E-1') Logarithm of factorials: `\log(x!)` ------------------------------------------------------------------------------- .. method:: math53.logfactorial(n) Returns `\log(x!) = \log(\Gamma(x+1))`. See also Wikipedia :cite:p:`WikipediaFun70`, MathWorld :cite:p:`WolframFun70`, BoostMath :cite:p:`BoostFun70`, :cite:t:`Ehrhardt2018` (3.5.4.3). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.LogFactorial(3) ereal('5.2359877559829887307E-1') >>> ereal.LogFactorial('0.51') ereal('5.3518479027559984754E-1') Logarithm of the binomial coefficient ------------------------------------------------------------------------------- .. method:: math53.logbinomial(n,k) Returns the logarithm of the binomial coefficient, `\displaystyle \log{n \choose k} = \log\left(\frac{n!}{k!(n-k)!}\right)\,`, for `n\ge k \ge 0`. See also Wikipedia :cite:p:`WikipediaFun72`, MathWorld :cite:p:`WolframFun72`, NIST :cite:p:`DLMFun72`, BoostMath :cite:p:`BoostFun72`, :cite:t:`Ehrhardt2018` (3.5.4.5). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.LogBinomial(13, 7) ereal('5.2359877559829887307E-1') >>> ereal.LogBinomial(12.6, '4.51') ereal('5.3518479027559984754E-1') .. _rst_log_beta: Log-Beta function ------------------------------------------------------------------------------- .. method:: math53.logbeta(a, b) where ``ctx`` is ``math53``, ``ctxcpp``, ``ctxboost`` or ``ctxflint``. Returns the logarithm of `B(a,b)` See also Wikipedia :cite:p:`WikipediaFun78`, MathWorld :cite:p:`WolframFun78`, NIST :cite:p:`DLMFun78`, BoostMath :cite:p:`BoostFun78`, :cite:t:`Ehrhardt2018` (3.5.3.2), Mpmath :cite:p:`MpmathFun78`. An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.LogBeta(3.1, 0.5) ereal('5.2359877559829887307E-1') >>> ereal.LogBeta(3.4, '0.51') ereal('5.3518479027559984754E-1') Inverse of the gamma function, `\Gamma^{-1}(y)` ------------------------------------------------------------------------------- .. method:: math53.gamma_inv(y) Returns `\Gamma^{-1}(y)`, the functional inverse of the gamma function, i.e. it returns `x` with `\Gamma(x)=y, \, y \ge 0.8857421875`. See also Wikipedia :cite:p:`WikipediaFun75`, MathWorld :cite:p:`WolframFun75`, NIST :cite:p:`DLMFun75`, BoostMath :cite:p:`BoostFun75`, :cite:t:`Ehrhardt2018` (3.5.1.3). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.GammaInv(1.5) ereal('5.2359877559829887307E-1') >>> ereal.GammaInv('1.51') ereal('5.3518479027559984754E-1') Inverse of the logarithm of the gamma function, `\log\Gamma^{-1}(y)` ------------------------------------------------------------------------------- .. method:: math53.lgamma_inv(y) Returns the functional inverse of `\log\Gamma(x)`, i.e. it returns `x = \log\Gamma^{-1}(y)` with `\log\Gamma(x) = y` for `y \ge -0.12142 > y_m` (the minimum of `\log\Gamma(x)` for positive arguments). The result is greater than `x_m = 1.46163\ldots` (the positive zero of the `\psi` function). See also Wikipedia :cite:p:`WikipediaFun77`, MathWorld :cite:p:`WolframFun77`, BoostMath :cite:p:`BoostFun77`, :cite:t:`Ehrhardt2018` (3.5.1.6). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.LogGammaInv(1.5) ereal('5.2359877559829887307E-1') >>> ereal.LogGammaInv('1.51') ereal('5.3518479027559984754E-1') Temme's regulated gamma function, `\Gamma^{*}(x)` ------------------------------------------------------------------------------- .. method:: math53.gammastar(x) Returns Temme's `\Gamma^{*}(x)`, defined by `\Gamma(x) = \sqrt{2\pi} e^{-x} x^{x-1/2} \Gamma^{*}(x)`. See also Wikipedia :cite:p:`WikipediaFun75`, MathWorld :cite:p:`WolframFun75`, NIST :cite:p:`DLMFun75`, BoostMath :cite:p:`BoostFun75`, :cite:t:`Ehrhardt2018` (3.5.1.4). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.GammaStar(1.5) ereal('5.2359877559829887307E-1') >>> ereal.GammaStar('1.51') ereal('5.3518479027559984754E-1') Relative Pochhammer symbol, `((a)_x - 1)/x` ------------------------------------------------------------------------------- .. method:: math53.poch1(a,x) Returns `\displaystyle \frac{(a)_x - 1}{x}`, accurate also for small `|x|`. For `x=0` the value `\psi(a)` is returned. See also :cite:t:`Ehrhardt2018` (3.5.4.7). An example in Python .. code-block:: pycon >>> from xlcalcnet import ereal >>> ereal.Poch1(13, 7) ereal('5.2359877559829887307E-1') >>> ereal.Poch1(12.6, '4.51') ereal('5.3518479027559984754E-1') .. _rst_mpm_catalan: Catalan function `C(x)` ------------------------------------------------------------------------------- .. method:: math53.catalan_c(x) where ``ctx`` is ``math53``, ``ctxcpp``, ``ctxboost`` or ``ctxflint``. Returns the Catalan function `\displaystyle C(x) = \frac{1}{x+1} \binom{2x}{x} = \frac{\Gamma(2x+1)}{(x+1)\Gamma(x+1)^2}`. See also: MathWorld :cite:p:`WolframFun301`, Wikipedia :cite:p:`WikipediaFun301`, :cite:t:`Ehrhardt2018` (3.10.5). An example with real input: .. code-block:: pycon >>> from xlcalcnet import dec, mpm, ipm >>> mpm.dps = 40; x = '10.5' >>> \mathrm{d}x = dec.catalan(x); mx = mpm.catalan(x); ix = ipm.catalan(x) >>> mpm.show([\mathrm{d}x, mx, ix]) dec: 3.137576033650317681318411712507890972194E+4 mpm: 3.137576033650317681318411712507890972194e+4 ipm: 3.137576033650317681318411712507890972194e+4 (3.597e-39%) >>> from xlcalcnet import mpm, fpm, gmp, apm >>> mpm.dps = 40; x = '10.5' >>> fx = fpm.catalan(x); gx = gmp.catalan(x); ax = apm.catalan(x) >>> mpm.show([fx, gx, ax]) fpm: 3.13757603365032E+04 gmp: 3.137576033650317681318411712507890972194E+04 apm: 3.137576033650317681318411712507890972194e+4 (3.597e-39%)