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digamma

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Digamma - Wikipedia
Digamma or wau (uppercase: Ϝ, lowercase: ϝ, numeral: ϛ) is an archaic letter of the Greek alphabet. It originally stood for the sound /w/ but it has remained ... en.wikipedia.org
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Digamma function - Wikipedia
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function. en.wikipedia.org
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Evidence for Digamma | Dickinson College Commentaries
To them, therefore, the digamma was little more than a symbol—the unknown cause of a series of metrical anomalies. In the present state of etymological ... dcc.dickinson.edu
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digamma
digamma (daɪˈgæmə) [a. L. digamma, Gr. δίγαµµα the digamma, f. δι- twice + γάµµα the letter gamma: so called by the grammarians of the first century, from its shape ϝ or F, resembling two gammas (Γ) set one above the other.] The sixth letter of the original Greek alphabet, corresponding to the Semit... Oxford English Dictionary
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Digamma : r/AncientGreek - Reddit
Digamma was the original Sixth letter of the Greek Alphabet. It was an F, but was dropped and the sound is preserved with Upsilon. www.reddit.com
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Digamma Function -- from Wolfram MathWorld
The digamma function is a special function which is given by the logarithmic derivative of the gamma function (or, depending on the definition, the logarithmic ... mathworld.wolfram.com
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Digamma function
The digamma function is often denoted as or (the uppercase form of the archaic Greek consonant digamma meaning double-gamma). Special values The digamma function has values in closed form for rational numbers, as a result of Gauss's digamma theorem. wikipedia.org
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The Use of Gamma in Place of Digamma in Ancient Greek in - Brill
This paper focusses on the use of gamma to replace the letter digamma, and to represent the /w/ sound in general. brill.com
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DIGAMMA Definition & Meaning - Merriam-Webster
1. a letter of the original Greek alphabet representing a sound approximately that of English w which early fell into disuse except in writing the western ... www.merriam-webster.com
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Digamma - Boost
Returns the digamma or psi function of x. Digamma is defined as the logarithmic derivative of the gamma function: The final Policy argument is optional and can ... www.boost.org
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scipy.special.digamma — SciPy v1.16.1 Manual
The digamma function. The logarithmic derivative of the gamma function evaluated at z. z array_like out ndarray, optional digamma scalar or ndarray docs.scipy.org
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25.4 Digamma | Stan Functions Reference
The digamma function Ψ Ψ is the derivative of the logΓ log Γ function, Ψ(u) = ddulogΓ(u) = 1Γ(u) dduΓ(u). mc-stan.org
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Series involving Digamma relates to Exponential Integral I came across the following series involving the Digamma function $\Psi$: \begin{equation} \sum^{\infty}_{k=0} \Psi(k+1) \frac{z^k}{k!}, \end{equation} where z...
Here is a derivation based on $$\psi(k+1)=-\gamma+\sum_{n=1}^{k}\frac{1}{n}=-\gamma+\int_0^1\frac{1-t^k}{1-t}\,dt$$ (let me use the conventional notation) and the formula $$\gamma=\int_0^1\frac{1-e^{-x}}{x}\,dx-\int_1^\infty\frac{e^{-x}}{x}\,dx.$$ We get \begin{align}\sum_{k=0}^{\infty}\psi(k+1)\fra...
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A property of the _digamma_ function On the last paragraph of the article on Wikipedia about the digamma function I found 1 _The digamma function appears in the regularization of divergent integrals_ $$ \int _{0}^{\i...
The idea of a regularization is to subtract some (constant) infinity to be left with a finite expression possibly meaning something. In this case, the relevant formula is mentioned in the article at Wikipedia: $$\psi (a)=-\gamma +\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}-{\frac {1}{n+a}}\right).$$...
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Digamma series Representation I was reading about where the series representation of digamma is proved and its states: $$-\frac{1}{x}-\gamma +\sum_{n=1}^{+\infty}\left(\frac{1}{n}-\frac{1}{n+x}\right) = -\gamma +\sum...
A good starting point is the Weierstrass product for the $\Gamma$ function: $$ \Gamma(x) = \frac{e^{-\gamma x}}{x}\prod_{n\geq 1}\left(1+\frac{x}{n}\right)^{-1}e^{x/n}\tag{1} $$ By considering $\frac{d}{dx}\log(\cdot)$ of both sides, $$ \psi(x) = -\frac{1}{x}-\gamma+\sum_{n\geq 1}\frac{x}{n(n+x)}=-\...
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