Digamma |
The digamma function is defined as the logarithmic derivative of the gamma function.
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GammaGergoNemes |
The Gergo Nemes' algorithm is very simple and quick to compute the Gamma function, if accuracy is not critical.
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GammaLanczos |
Lanczos approximation provides a way to compute the Gamma function such that the accuracy can be made arbitrarily precise.
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GammaLanczosQuick |
Lanczos approximation, computations are done in double .
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GammaLowerIncomplete |
The Lower Incomplete Gamma function is defined as:
\[
\gamma(s,x) = \int_0^x t^{s-1}\,e^{-t}\,{\rm d}t = P(s,x)\Gamma(s)
\]
P(s,x) is the Regularized Incomplete Gamma P function.
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GammaRegularizedP |
The Regularized Incomplete Gamma P function is defined as:
\[
P(s,x) = \frac{\gamma(s,x)}{\Gamma(s)} = 1 - Q(s,x), s \geq 0, x \geq 0
\]
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GammaRegularizedPInverse |
The inverse of the Regularized Incomplete Gamma P function is defined as:
\[
x = P^{-1}(s,u), 0 \geq u \geq 1
\]
When s > 1 , we use the asymptotic inversion method.
When s <= 1 , we use an approximation of P(s,x) together with a higher-order Newton like method.
In both cases, the estimated value is then improved using Halley's method, c.f., HalleyRoot .
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GammaRegularizedQ |
The Regularized Incomplete Gamma Q function is defined as:
\[
Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}=1-P(s,x), s \geq 0, x \geq 0
\]
The algorithm used for computing the regularized incomplete Gamma Q function depends on the values of s and x.
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GammaUpperIncomplete |
The Upper Incomplete Gamma function is defined as:
\[
\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t = Q(s,x) \times \Gamma(s)
\]
The integrand has the same form as the Gamma function, but the lower limit of the integration is a variable.
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Lanczos |
The Lanczos approximation is a method for computing the Gamma function numerically, published by Cornelius Lanczos in 1964.
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LogGamma |
The log-Gamma function, \(\log (\Gamma(z))\), for positive real numbers, is the log of the Gamma function.
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Trigamma |
The trigamma function is defined as the logarithmic derivative of the digamma function.
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