Questions tagged [digamma-function]

The digamma function, usually represented by the Greek letter psi or digamma, is the logarithmic derivative of the [tag:gamma-function]. It is the first of the polygamma functions.

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3
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1answer
116 views

Is $\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n+1}$a complex number? What is happening?

While computing the integral $$\displaystyle\int_0^1{\displaystyle\sum_{n=1}^{\infty}x^{(2^n)}dx}$$ I easily got to $$\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n+1}$$ Since this was getting ...
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0answers
60 views

Find a function $\phi(a)$ which fixes the area under $f(x)=x^a e^{-\phi(a)x}$ for all $a>0$

Find a function $\phi:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ which fixes the area under $f(x)=x^a e^{-\phi(a)x}$ for all $a>0$. My investigation so far has lead me to believe that no such ...
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Is there a more elementary way to arrive at: $\ln(m)=\lim_{n\to\infty}\sum_{k=n+1}^{mn}\frac{1}{k}$?

Let $\psi$ denote the digamma function, and let $m,n\in\Bbb N_1$. For $m=1$, use the empty sum as $\ln(1)=0$ anyway! The Gauss multiplication tells us that: $$\sum_{i=1}^{m-1}\psi\left(n+\frac{i}{m}\...
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0answers
47 views

Simplification of a difficult identity involving the digamma function

EDIT: The question here is not for the reader to laboriously scan all the working, but rather to suggest ways to continue the train of thought; for example, there is perhaps a closed form of the ...
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1answer
33 views

Use of digamma function to evaluate some finite series

I need to evaluate a series of a function that switches sign in the following way: \begin{eqnarray} \sum_{k=-\infty}^{+\infty}\frac{\text{sgn}(n-k)}{((2n+1)+B\text{sgn}(n-k)) -(2k+1)} \end{eqnarray} ...
4
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3answers
215 views

$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$

$$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$$ I tried to solve it by using this product formula, $$\frac 1{\Gamma (x)}=xe^{\gamma x} \prod_{n=...
5
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1answer
239 views

Mistake computing $\int_0^\infty\frac{x \ln(1+x^2)}{\sinh \pi x}\,dx=\frac{\ln 2}{3} - \log \pi - \frac{1}{2} + 6 \ln A $

Edit I found the mistake, see my answer below. I am trying to evaluate the integral $$\int_0^\infty\frac{x \ln(1+x^2)}{\sinh \pi x}\,dx=\frac{\ln 2}{3} - \log \pi - \frac{1}{2} + 6 \ln A $$ I know ...
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Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions

It is common knowledge that the classical Euler gamma function $\Gamma(z)$ can defined by \begin{equation*} \Gamma(z)=\int^\infty_0t^{z-1} e^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and the ...
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2answers
206 views

Computing $\sum_{n=2}^{\infty}\frac{1}{(n^2-1)^2}=\frac{\pi^2}{12}-\frac{11}{16}$

I want to compute the following infinite sum $$\sum_{n=2}^{\infty}\frac{1}{(n^2-1)^2}=\frac{\pi^2}{12}-\frac{11}{16}$$ To this goal, my strategy is to first compute $\sum_{n=2}^{\infty}\frac{1}{n^2-a^...
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1answer
62 views

What's the easiest way to derive an approximation of $ \frac{d}{dz} \log \Gamma(z) $ from the literature?

My main problem is with the error term. I tried to start with Stirling's formula in the form $$ \log\Gamma(z) = ... + \mu(z) $$ with the holomorphic function $$ \mu(z) := - \int_0^\infty \frac{P(t)}{z+...
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1answer
45 views

Show an inequality about digamma function

Let $\psi$ be the digamma function, i.e. $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$. Show that for $x>0$, \begin{equation} \label{1} \psi\left(1+\frac{1}{2x}\right)-\psi\left(\frac{1}{2}+\frac{1}{2x}\...
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0answers
181 views

Evaluating an infinite series containing Digamma Function

I was evaluating the integral $$\mathcal{I}=\int_0^1 \frac{\ln^2 x\ln(1-x)}{x^2+1}\mathrm dx$$ I tried to use the series $$\ln(1-x)=-\sum_{n=1}^\infty \frac{x^n}{n}$$ and interchanged the sum and ...
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2answers
290 views

Sum with Digamma function expressed by hypergeometric function?

The function $$f(x)=\sum_ {k = 0}^{\infty} \frac {(-1)^ k x^{2 k}} {2^k k! (2 k + 2)! }\left(\psi (k + 2)+ \frac{1}{2}\psi\left(k + \frac{3}{2}\right)\right)\tag{1}$$ with $x>0, x\in\mathbb{R}$ ...
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2answers
107 views

Sum a product of Bernoulli numbers and binomial coefficients

Context: I am interested in developing the large-$x$ asymptotic series of the digamma function $$\psi\Big(\frac{1}{2}+ix\Big)$$ for real positive $x$. For this I am using the known asymptotic ...
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1answer
101 views

Showing the existence of the Euler-Mascheroni constant

I want to show that there exists a constant $\gamma$ such that: $$- \frac{\Gamma'(z)}{\Gamma(z)} = \gamma + \frac{1}{z} + \sum_{n=1}^\infty \left(\frac{1}{z+n}-\frac{1}{n} \right)$$ where $\Gamma$ is ...
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5answers
130 views

Approximating the Digamma function for small arguments

There are several ways to approximate the Digamma function $\psi(x)$ that become exact for $x\to\infty$. The simplest approximation is $$\lim_{x\to\infty}(\psi(x)-\mathrm{ln}(x))=0$$ There are other ...
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36 views

What is the minimum of indefinite sum of $\tan x$?

The indefinite sum of $\tan x$ is the following function: $T(z)=-\sum _{k=1}^{\infty } \left(\psi \left(k \pi -\frac{\pi }{2}+1-z\right)+\psi \left(k \pi -\frac{\pi }{2}+z\right)-\psi \left(k \pi -\...
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1answer
37 views

How do you derive the "reflection formula" for the digamma function?

In a question at CrossValidated we have a user asking for derivation of the "reflection formula" for the digamma function. Using integers $0 \leqslant x \leqslant n$ the formula of interest ...
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198 views

Compute the double sum $\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)$

I am trying to compute the following double sum $$\boxed{\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)}$$ I proceeded as following $$\sum_{n=1}^{\infty}\sum_{m=1}^...
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6answers
318 views

Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(2n+1)}=2-2\ln(2)$

I came across the following statements $$\sum_{n=1}^{\infty} \frac{1}{n(2 n+1)}=2-2\ln 2 \qquad \tag{1}$$ $$\sum_{n=1}^{\infty} \frac{1}{n(3 n+1)}=3-\frac{3 \ln 3}{2}-\frac{\pi}{2 \sqrt{3}} \qquad \...
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2answers
237 views

How to write a limit in terms of finite summation

I managed to find$$\int\limits_0^\infty \frac{\ln^{2a}(x)\ln(1+x)}{\sqrt{x}(1+x)}\mathrm{d}x=-\pi\lim_{m\to \frac12 }\frac{\mathrm{d}^{2a}}{\mathrm{d} m^{2a}} \frac{\psi(1-m) + \gamma}{\sin(m\pi)}.$$ ...
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1answer
44 views

How do you interpret the infinite activity property of variance gamma process?

I am a little bit confused with the infinite activity property of variance gamma(VG) process $X(t),$ where $$X(t)=\theta G(t) + \sigma W(G(t)),$$ for any finite interval, the VG process has infinitely ...
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1answer
65 views

Show $\Psi\left(n + \frac{1}{2}\right) - \Psi(n + 1) = -2\int_0^1\frac{x^{2n}}{1 + x}\,dx $

Let $ \Psi(n) $ denote the Digamma function, how to prove that $$\Psi\left(n + \dfrac{1}{2}\right) - \Psi(n + 1) = -2\int_0^1\dfrac{x^{2n}}{1 + x}\,dx $$ I have tried using the integral ...
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1answer
22 views

exponential distribution and the gamma function

I don't find the relationship between the expected value (theoretical) of an exponential distributed variable and the gamma function. I work on the paper Moments of the Log ACD model from Luc Bauwens(...
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0answers
47 views

Complex Limit containing gamma and digamma function:

Consider the following function : $$f(x)=\frac{x\Gamma'(x)}{(\Gamma(x))^2}-\frac{1}{\Gamma(x)}$$ I need to calculate the following limit: $$\lim_{y\rightarrow\infty}\frac {|f(x+iy)|}{e^{2πy}}$$ I '...
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1answer
356 views

Beautiful monster: Catalan's constant and the Digamma function

The problem I have been trying for a while now to show that this monster $$\begin{align} &\int_0^{\pi/4}\tan(x)\sum_{n=1}^{\infty}(-1)^{n-1}\left(\psi\left(\frac{n}{2}\right)-\psi\left(\frac{n+1}{...
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3answers
195 views

Integration using digamma functions

I'm trying to prove that for $x,y,r>0$ the following identity (gotten via CAS) holds: $$\int_0^x \frac{x^r}{x^r+y^r}\,dy=\frac{x}{2r}\left(\psi\left(\frac{1+r}{2r}\right)-\psi\left(\frac{1}{2r}\...
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4answers
190 views

Closed form of $\sum_{m=1}^{\infty} \frac{(-1)^mH_{\frac{3m}{4}}}{3m}$

I've been working on an integral, namely: $${\displaystyle \int_0^1 \frac{x^2}{1 + x^3}\ln(1 - x^4)dx}$$ Which I managed to narrow down to the following expression: $$\sum_{m=1}^{\infty} \frac{(-1)^...
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1answer
40 views

Monotony of $x\psi(x)$ and $x^2\psi(x)$ on the positive reals.

Are $f(x)=x\psi(x)$ and $g(x)=x^2\psi(x)$ monotonically increasing on $x\in(0,\infty)$? I found $f^\prime(x)=\psi(x)+x\psi^{(1)}(x)$ and $g^\prime(x)=2x\psi(x)+x^2\psi^{(1)}(x)$ so if I could show ...
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1answer
65 views

prove an inequality containing digamma function

I wonder how to prove this or if it is correct since that from numerical result, it is correct $$ (2 \log 2-1) x^{2}+(2 \log 2+2) x+2+\left(2 x^{2}+2 x\right) \Psi(x)>0 $$ where $x\in(0,+\infty)$
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1answer
42 views

a limit containing a digamma function

I wonder if the following limit is correct and how to prove it $$ \lim _{x \rightarrow+\infty} \frac{4}{p^{2}} x^{4}\left(\frac{1}{x^{2}}+\frac{2}{x}(\log 2+\Psi(x))+\Psi^{\prime}(x)\right)>0 $$ ...
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0answers
56 views

Taylor expansion of the reciprocal gamma function

with regard to the post Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function, now consider the function $$ f(x)=\frac{1}{[\Gamma(x)]^n}, $$ where $n\in \...
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2answers
99 views

Simplify this expression $e^{\psi\left(\frac12+\frac{i}{2\sqrt{3}}\right)+\psi\left(\frac12-\frac{i}{2\sqrt{3}}\right)}$?

Is it possible to simplify this constant expression $e^{\psi\left(\frac12+\frac{i}{2\sqrt{3}}\right)+\psi\left(\frac12-\frac{i}{2\sqrt{3}}\right)}$? Here $\psi(x)$ is digamma function. Particularly, ...
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2answers
210 views

Asymptotic Expansion of Digamma Function

While reading the wikipedia page of the Digamma function (https://en.wikipedia.org/wiki/Digamma_function#Asymptotic_expansion) I noticed that it said the asymptotic expansion for the digamma function (...
6
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3answers
225 views

Evaluating $I(z,s)=\int_0^1\int_0^1\left(1-\frac{(1-x)(1-y)}{(1-(1-z)x)(1-(1-z)y)}\right)^{s-2}\,\mathrm dx\mathrm dy$

I came across the following double integral in a statistics problem: For $z>0$ $$ I(z,s)=\int_0^1\int_0^1\left(1-\frac{(1-x)(1-y)}{(1-(1-z)x)(1-(1-z)y)}\right)^{s-2}\,\mathrm dx\mathrm dy. $$ All I ...
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1answer
73 views

Digamma function Identities

I encountered following expression $$\psi ^{(0)}\left(z+\frac{1}{4}\right)-\psi ^{(0)}\left(z-\frac{1}{4}\right)$$ Searching in different resources I managed to find these identities involving the ...
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1answer
38 views

Derivation step in paper on LDA

I am wondering how do the authors of this paper perform the following step (in the paper it is on page 29): $$\dfrac{\partial L}{\partial \gamma_i}=\Psi^\prime(\gamma_i)\left(\alpha_i + \sum_{n=1}^N\...
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0answers
48 views

Using calculus to find the maximum involving digamma function (in Latent Dirichlet Allocation).

I am having difficulty seeing how the authors (in Appendix A.3.2 under "Variational Dirichlet" of this paper) maximise the function $L$ with respect to $\gamma_i$ to derive a solution for $\...
3
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1answer
66 views

Why does $\operatorname{Im}\psi\left(\frac{1}{2}+i x\right)=\frac{\pi}{2}\tanh(\pi x)$, and what can be said of the real part?

Wikipedia claims that $$\operatorname{Im}\psi\left(\frac{1}{2}+i x\right)=\frac{\pi}{2}\tanh(\pi x),$$ where $\psi$ is the digamma function. How can we prove this identity? Does a similar identity ...
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2answers
249 views

Closed form of the sum $\sum_{n=1}^{\infty}\frac{H_n}{n^x}$

Some days ago I derived the identity $$\sum_{n=1}^{\infty}\frac{H_n}{n^2}=2\zeta(3)$$ where $H_n$ is the $n$th Harmonic number. Other related identities include $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\...
4
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2answers
107 views

A convergent series for the Trigamma function $\psi_1(n) =\sum_{k=n}^{\infty} \frac1{k^2} $

I just came up with the following convergent series for the Trigamma function defined by $\psi_1(n) =\sum_{k=n}^{\infty} \frac1{k^2} $. \begin{align*} \psi_1(n) &=\lim_{m \to \infty} \sum_{j=1}^m \...
5
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2answers
230 views

Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$

Note:This is the same question, but it doesn't answer my question, the answer doesn't give a closed form. In fact, the answer is not accepted. Moreover, I don't think that a 9 month old inactive ...
0
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1answer
44 views

Why does $\frac{d}{d\nu}\ln\Gamma\left(\frac{\nu}{2}\right)$ simplify to digamma function?

$$-(\nu+1)\frac{d}{d\nu}\left[\ln\Gamma\left(\frac{\nu}{2}\right)-\ln\Gamma\left(\frac{\nu+1}{2}\right)\right]=\frac{\nu+1}{2}\left[\psi\left(\frac{\nu+1}{2}\right)-\psi\left(\frac{\nu}{2}\right)\...
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1answer
96 views

Deriving closed form of $\sum_{n=1}^\infty\frac{(-1)^{n+1}\ln(n+1)}{n} $

I wanted to derive closed form of $\sum_{n=1}^\infty\frac{(-1)^{n+1}\ln(n+1)}{n}$ which I converted into a integral, $$\sum_{n=1}^\infty\frac{(-1)^{n+1}\ln(n+1)}{n}=\int_0^1\int_0^1\frac{u^v}{1+u^v}\...
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1answer
80 views

Dimension over $\mathbb{Q}$ of infinite sums of rational functions

Let $P(n)=(n+r_1)(n+r_2)...(n+r_k)$ be a polynomial with simple, rational, negative roots (i.e. $r_i>0$) and degree $k\geq 2$ (I stick with negative roots as I don't have to worry about dividing by ...
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1answer
101 views

Does there exist a formula for $\int_0^{\infty} t^{k} {\tt sech}(t)dt$ that is correct whenever the real part of k is greater than negative 1?

The formula $\int_0^{\infty} t^{k} {\tt sech}(t)dt=\frac{(-1)^k}{2^{2k+1}} \left( \psi^{(k) } \left( \frac {3} {4} \right) -\psi^{(k)}\left( \frac {1} {4} \right) \right) ...
0
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1answer
43 views

Is $\psi(x)-\log x$ strictly increasing for strictly positive $x$?

Let $\psi(x)$ be the digamma function. Is the function which takes $\psi(x)-\log x$ for $x>0$ strictly increasing, and how could one show this if it is the case (link etc.)?
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0answers
44 views

Finite sum of reciprocals of odd integers in terms of digamma function

I was reading some of my old notebooks and I came across an (according to me) astounding formula: $$\sum_{k=1}^{n}\frac{1}{2k-1}=\frac{\psi(n+\frac{1}{2})+\gamma}{2}+\log 2$$ I almost never gave ...
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1answer
47 views

Derivative of Bessel $K$

I'm interested in the first derivative of the Bessel $K$ function with respect to its parameter. I'll use the following notation; $K^{(1,0)}(n,z):= \frac{\partial}{\partial \nu} K_\nu(z) \bigg|_{\nu = ...
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1answer
60 views

Is there a decomposition for the digamma function as a sum of digamma functions?

Let $\psi(x)$ denote the digamma function $$ \psi(x)=\Gamma(x)\frac{\partial}{\partial x} \Gamma(x). $$ Consider $x=x_1 +x_2+\dots +x_m$, where $x_j>0$, for $j=1, \ldots,m$. Is there any formula to ...