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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Sum of over primes $p$ of $x^{-p}$

I was playing around with the following series $$S(x) = \frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+...=\sum_{p\in primes}\frac{1}{x^p}$$ for $x\in\mathbb{R}$ and $|x|>...
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Deriving asymptotic for the roots of Digamma Function

So Wikipedia gave these asymptotics for the Digamma function: $$x_n=-n+\frac12+O\left(\frac1{(\ln n)^2}\right)$$$$x_n\approx-n+\frac1\pi\arctan\left(\frac{\pi}{\ln n}\right)$$$$x_n\approx-n+\frac1\pi\...
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Proving that $\frac{y}{2x(y + x)} - \log(x) + \log(y + x) + \psi(x) - \psi(y + x) \leq 0$ for $x \geq 1, y > 0$

I am trying to evaluate a particular approximation to a distribution function. As a subproblem, I am trying to prove that for all $x \geq 1, y > 0$ we have: $$ \frac{y}{2x(y + x)} - \log(x) + \log(...
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Prove that $\lim_{x\to-\infty}\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-\ln 2\right)\sin(\pi x)=\pi$

I was messing around on Desmos when I saw this interesting limit: $$\lim_{x\to-\infty}\left(\psi(x)-\psi\left(\frac x2\right)-\frac1x-\ln 2\right)\sin(\pi x)=\pi$$This is of course what I think is the ...
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Another weird limit involving gamma and digamma function via continued fraction

Context : I want to find a closed form to : $$\lim_{x\to 0}\left(\frac{f(x)}{f(0)}\right)^{\frac{1}{x}}=L,f(x)=\left(\frac{1}{1+x}\right)!×\left(\frac{1}{1+\frac{1}{1+x}}\right)!\cdots$$ Some ...
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Dimension of maximum volumed unit ball

Let $V_n=V(B^n)$ be the volume of the $n$-dimensional unit ball $B^n$. By cross-sectioning $B^n$ along $x_n$-axis, $-1\leq x_n\leq 1$ and by means of similarity of hyper disks we have $$V_n=2\int_0^1(\...
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Prove $\frac12\left(\psi\left(\frac{x+1}2\right)-\psi\left(\frac x2\right)\right)=\psi(x)-\psi\left(\frac x2\right)-\ln2$

Desmos suggests that$$\frac12\left(\psi\left(\frac{x+1}2\right)-\psi\left(\frac x2\right)\right)=\psi(x)-\psi\left(\frac x2\right)-\ln2$$Where $\psi$ is the digamma function. I can write the LHS as $$\...
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Digamma function property

I am trying to determine $\displaystyle\sum_{n=0}^{\infty}\dfrac{1}{(\alpha\cdot n+1)^2} $, to do this I will use the following property that I cannot prove, any ideas? $$ \sum_{n=0}^\infty\frac1{(\...
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Estimating derivative involving the gamma function

Let \begin{align} \lambda'(p)=-c\frac{1}{(p+2)^\frac{3}{2}}\frac{\Gamma\left(\frac{3p}{4}-\frac{1}{2}\right)}{\Gamma\left(\frac{3p}{4}\right)}\left(\frac{3}{2(p+2)}+\frac{3}{4}\left(\psi(\frac{3p}{...
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Digamma equation for fitting

I have a digamma equation: ln(1/t) = psi(1/2 + h/2t) - psi(1/2). I would like to reorganize this equation to apply for experimental data fitting (h and t are data points, t is input and h is output). ...
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Function with sum equals function which uses digamma function

currently I am working on a Formular which I want to maximize. I tried to simplify the function with Wolfram. The Result was the following: $f(k)=\sum_{i=k}^{n} \left( \frac{k-1}{(i-1)n} + \frac{(k-1)...
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Convexity of a scaled multivariate digamma function

The problem... Let $\psi_p(a) = \frac{\partial \Gamma_p(a)}{\partial a}$ be the multivariate digamma function. I believe the following function to be strictly convex at least for real values $a>2p$...
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Do we have $\log\Gamma(2x+1)-2\log\Gamma(x+1)\ge\log(x^2+1)$ for $x\in[0,1]$?

Prove or disprove that, for all $x \in [0, 1]$, $$\log\Gamma(2x+1)-2\log\Gamma(x+1)\ge\log(x^2+1).$$ Here $\Gamma$ is the gamma function. I discovered this relation by accidentally drawing the graph ...
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About the integral $\int_0^{\pi/2} (\sin x-1)/\ln(\sin x) \mathrm{d}x$

Using the Feynman Technique, Let $$I(a) := \int_0^{\pi/2} \frac{(\sin x)^a - 1}{\ln(\sin x)} \mathrm{d}x$$ $$I’(a)= \int_0^{\pi/2} (\sin x)^a \mathrm{d}x = \frac{\sqrt{π}}{2} \frac{\Gamma\left(\...
integral's user avatar
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Series expansion for $\displaystyle{\psi(t+2)-\psi\left(\frac{t+3}{2}\right)}$

I need the power series (starting with index $k=1$) of a difference between two digamma functions. $$\psi(t+2)-\psi\left(\frac{t+3}{2}\right)=\sum_{k=1}^\infty a_kt^k$$ In otherwords I want $a_k$ in ...
bob's user avatar
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How to finish the integral $\int_{0}^{\infty}\frac{\sin(x)}{\sinh(x)}\,dx$ [duplicate]

I was evaluating: $$I:=\int_{0}^{\infty}\frac{\sin(x)}{\sinh(x)}\,dx$$ This is what I did, how can my answer be simplified if correct. The following is my work: $$I=2\int_{0}^{\infty}\frac{\sin(x)}{e^...
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What are some formulas involving Dottie number and digamma function?

Here are two formulas that involve Dottie number and Gamma function: $\Gamma \left( \frac{1}{2} + \frac{d}{\pi} \right) \Gamma \left( \frac{1}{2} - \frac{d}{\pi} \right) = \frac{\pi}{d} $ and $\sqrt{\...
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Does $\sum_{r = 1}^n \ln\left(\frac{1 + r}{r}\right) = \ln (\Gamma(n + 2)) - \ln (\Gamma(n+1))$? If so, why?

When attempting the evaluate the integral $\int_0^1 \{\ln(x)\}$, where $\{ x \}$ is the fractional part function, I came across the following sum: $$\sum_{r = 1}^n \ln\left(\frac{r + 1}{r}\right) $$ ...
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What is the intersection point of the graphs of digamma and trigamma functions?

We have the series expansions of digamma and trigamma functions for $x>0$, $$\psi^{(0)}(x)=-\gamma-\frac1x+\sum_{k=1}^{\infty}(\frac1k-\frac1{x+k})$$ and $$\psi^{(1)}(x)=\sum_{k=0}^{\infty}\frac{1}{...
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Showing $\displaystyle{\frac{\psi(z)}{\Gamma(z)}=-e^{2\gamma z}\prod_{k=0}^{\infty}\left(1-\frac{z}{x_k}\right)e^{z/x_k}}$

I want to show the following infinite product, $$\frac{\psi(z)}{\Gamma(z)}=-e^{2\gamma z}\prod_{k=0}^{\infty}\left(1-\frac{z}{x_k}\right)e^{z/x_k}$$ where $x_k$ is the $k$th root of the digamma ...
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Proving $\sum_{k=1}^\infty\left(\frac{(n^k+1)}{(n+1)^k}\zeta(k+1)\right)=\psi(\frac{n}{n+1})-\psi(\frac{1}{n+1})=\pi\cot(\frac{\pi}{n+1})$

I'm an amateur/hobbyist mathematician, and I found this interesting relationship about 6 years ago, but haven't ever quite understood it! I feel like this is related to how Digamma and Zeta are ...
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Harmonic and Digamma function

I am a master's student, I have a little experience in harmonic numbers. I know the $H_{n} = \sum_{k=1}^{n} \frac{1}{k}$ . I am wondering what the $H_{(\frac{1}{k}-1)}$ is. I am trying to rewrite it ...
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Having trouble evaluating an integral with logarithm and binomial expansion to difference of digamma functions

I'm trying to understand the derivation of the Kozachenko-Leonenko entropy estimator which uses a k-nearest neighbor approach to estimate a probability density function. The details of the derivation ...
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Finding the local extrema of $x^{\frac{x}{x!}}$

I want to find the local extrema of $x^{\frac{x}{x!}}$ for all $x>0$. Using Desmos, I got $(0.379,0.661)$ and $(2.228,2.046)$. What I want to do is find the exact values. Here is what I tried: $$f(...
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Closed form expression for an integral

Let $\psi_q(z)$ be the q-DiGamma function defined for a complex variable $z$ with $\Re(z)>0$ as $$\psi_q(z)=\frac{1}{\Gamma_q(z)}\frac{\partial}{\partial z} (\Gamma_q(z))$$ where $\Gamma_q(z)$ is ...
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What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$? [duplicate]

What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$? $F_n$ denotes the $n^{th}$ Fibonacci number. Wolframalpha gave me this answer: $$\sum_{n=1}^{\infty}\frac{1}{F_n}\ =\frac{1}{4}\...
Dylan Levine's user avatar
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Closed form expression for $\psi_{e^{\pi}}^{(3)}(1-i)$

Let $\psi_q(z)$ be the q-DiGamma function defined for a real variable $\Re(z)>0$ as $$\psi_q(z)=\frac{1}{\Gamma_q(z)}\frac{\partial}{\partial z} (\Gamma_q(z))$$ where $\Gamma_q(z)$ is the q-Gamma ...
Max's user avatar
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Closed form expression for $\psi_{e^{\pi}}^{(3)}(1)$

Let $\psi_q(x)$ be the q-DiGamma function defined for a real variable $x>0$ as $$\psi_q(x)=\frac{1}{\Gamma_q(x)}\frac{\partial}{\partial x} (\Gamma_q(x))$$ where $\Gamma_q(x)$ is the q-Gamma ...
Max's user avatar
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Inverse of difference of two digamma functions

I recently encountered the expression below for which I was interested in solving for $x$: \begin{equation} \psi(x+n+1) - \psi(x+1) =y \end{equation} $\psi$ is the digamma function, $n$ is a positive ...
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How to prove $\sum_{k=1}^n{n\choose k}\frac{(-1)^{k+1}}{k}=\sum_{k=1}^n\frac{1}{k}$?

I think by induction we can do it. Let $I(n)=\sum_{k=1}^n{n\choose k}\frac{(-1)^{k+1}}{k}=\sum_{k=1}^n\frac{1}{k}.$ Then, we must show that $I(n+1)-I(n)=\frac{1}{n+1}$. $\begin{align} I(n+1)-I(n)&...
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Gauss's Digamma Theorem generalization [closed]

This website shares a proof for Gauss' Digamma Theorem. Which is $$ ψ\left(\frac{p}{q}\right) = -\gamma - \frac{\pi}{2}\cot\left(\pi\frac{p}{q}\right) - \ln(q) + \frac{1}{2}\sum_{k=1}^{q-1} \cos\left(...
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What will be this limit? I'm having a hard time finding this....

What will be this limit? \begin{equation} \lim_{\ n\to\infty}\ \left(\frac{\left(n!\right)}{n}\right)^{\frac{1}{n}}=P\ \end{equation} I tried it like this: its in the form (infinity)^(0), so taking ...
math is magical's user avatar
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Seeking for other methods to evaluate $\int_0^{\infty} \frac{\ln \left(x^n+1\right)}{x^n+1} dx$ for $n\geq 2$.

Inspired by my post, I go further to investigate the general integral and find a formula for $$ I_n=\int_0^{\infty} \frac{\ln \left(x^n+1\right)}{x^n+1} dx =-\frac{\pi}{n} \csc \left(\frac{\pi}{n}\...
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Do we have a closed form for $\int_0^{\infty} \frac{\ln t}{\left(1+t^2\right)^n} d t $? [duplicate]

Latest Edit We are glad to see there are 4 alternative solutions which give the same closed form to the integral: $$\boxed{\int_0^{\infty} \frac{\ln t}{\left(a^2+t^2\right)^n} d t = \frac{a^{1-2n}\...
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Find reference about expansion of gamma function

If there are any reference about the expansion of $$\Gamma\left(\frac{p-x}{q}\right),$$ where $p,q$ are integers with $1\leq p\leq q$?
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Finding $\sum_{n=0}^{\infty} (-1)^n\left(\frac{1}{(3n+2)^2}-\frac{1}{(3n+1)^2}\right)$

Recently, I stumbled upon a summation $$S=\sum_{n=0}^{\infty} (-1)^n\left(\frac{1}{(3n+2)^2}-\frac{1}{(3n+1)^2}\right)$$ which can luckily be summed to a good number. Use $\psi^1(z)=\sum_{n=0}^{\infty}...
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Prove $(n-1)! + 1 = n^2$ has only one integer solution [duplicate]

Prove $(n-1)! + 1 = n^2$ has only one integer solution, namely $5$. I think that we can use the derivate of the gamma function to say that the LHS is growing more than the RHS from $n=5$ onwards, so ...
HappyFace's user avatar
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Computing solutions to polygamma function of order 1?

I want to preface this with the fact that I am WAY out of my depth with my mathematical familiarity with these topics. While trying to figure out how to compute the solution to the polygamma function ...
YaGoi Root's user avatar
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Derivation of Digamma function

In the paper by Kraskov et al (2004) there is a rather large jump in calculations. I am wondering if someone could fill out the gap for the equation below (equation 17 in the paper): $$ k\binom{N-1}{k}...
Anton's user avatar
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Switching the sum and integral

So I was calculating $\psi\left(\frac{1}{3}\right)$, I started with $$\psi(s+1)=-\gamma+\sum_{n=1}^{\infty}\frac{1}{n}-\frac{1}{n+s}$$ putting $s=-\frac{2}{3}$ and do a variable shift, $$\psi\left(\...
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Show the limit $\lim_{\delta \to 0}\int_{\ln(1+\delta)}^{\delta} \frac{e^{-x}}{x}\,dx \to 0 $ in a proof of the Digamma function

I want to show that $$\lim_{\delta \to 0}\int_{\ln(1+\delta)}^{\delta} \frac{e^{-x}}{x}\,dx \to 0 \tag{1}$$ Intutitively I could take the limits before integration, then I would get $$\int_{0}^{0} \...
Ricardo770's user avatar
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Bound this integral $\displaystyle\int_0^\infty \frac{(1-p+pe^{-t})^n -e^{-npt} -(1-p)^n(1-e^{-nt})}{1-e^{-t}}dt$ [closed]

If $n\in\mathbb N$ and $p\in (0,1)$, can the integral below be bounded by $\frac{1}{np}$ or $\frac{1}{np} +\mathcal O(n^{-1})$? $$\int_0^\infty \frac{(1-p+pe^{-t})^n -e^{-npt} -(1-p)^n(1-e^{-nt})}{1-e^...
Rodrigo Morales's user avatar
3 votes
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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 ...
Kinesis's user avatar
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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 ...
Gentleman_Narwhal's user avatar
1 vote
3 answers
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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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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 ...
FShrike's user avatar
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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} ...
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4 votes
3 answers
286 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=...
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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 ...
Ricardo770's user avatar
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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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