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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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Evaluation of a sum by means of Poisson sum formula and digamma function

I have the following series: $$\sum_{n=-\infty}^{\infty}\frac{1}{(2n+1)^2\pi^2+a^2}=\frac{1}{2a}\tanh\left(\frac{a}{2}\right)$$ and on the text it is written that it can be proven by means of either ...
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Prove $\gamma = \frac{1}{2} + 2 \cdot \int_0^\infty \frac{\sin(\arctan(x))}{(e^{2 \pi x} - 1) \cdot \sqrt{1 + x^2} } dx$

I've found the following integral on the Wikipediapage of the Euler-Mascheroni constant and I want to prove it. $\gamma = \frac{1}{2} + 2 \cdot \int_0^\infty \frac{\sin(\arctan(x))}{(e^{2 \pi x} - 1) ...
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digamma function inverse and special value

What the inverse of the digamma function?, and how can I write the x for $$ ψ(x)=1$$ and $$1<x$$ $[x ≈ 3.20317146837693106929448152]$ as irrasional number [not a new one a familiar old one]
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Application of Legendre's duplication formula

I am reading the book "Special functions: an introduction to classical functions of mathematical physics" by Nico M. Temme and I'm having trouble understanding how to find a constant (in page 62). I ...
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How to Prove Integral Representations of $H_z$ and $\psi(z)$

I've run across several integral representations of $H_z$ such as the following (see Harmonic Number Integral Representations). (1) $\quad H_z=\int\limits_0^1 \frac{1-t^z}{1-t} \, dt\,,\qquad\qquad\...
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Derivative of $\Gamma (1)=-\gamma$

How to compute $\Gamma'(1)=-\gamma$ where $\gamma$ is Euler's constant i-e the limit of the series $(1+\frac12 +\frac13 +\frac14 +...+\frac1n)-\ln(n)$ where n $\rightarrow \infty$ $\gamma= 0....
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Finding an Inverse of Restricted Gamma Function

I don't know/haven't used LaTeX yet but I'll do my best to keep it simple, I'm working on my undergrad senior project and I'm trying to find an inverse function for f(x)=(x-1)! just in the positive ...
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Can the following integral involving digamma be evaluated in closed form (special functions allowed but not preferred)?

$$\int_1^n \frac {\psi(x)}{x} dx \ s.t. \ n \in \mathbb N$$ EDIT: I've relocated the asking about the integral of the above except with the q-analog of the digamma to another question. This is so as ...
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Evaluate $\sum_{n=1}^{\infty}\frac{\psi^{''}(n)}{2n-1}$, where $\psi^{''}(n)$ is 2nd derivative of digamma function.

Does the following sum have a closed form? $$\sum_{n=1}^{\infty}\frac{\psi^{"}(n)}{2n-1},$$ where $\psi^{"}(n)$ is 2nd derivative of digamma function.
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Sums of Reciprocals of Polynomials and Harmonic Numbers

This is question based on a pattern I have noticed while using mathematica. Let $P(x)$ be a polynomial with real, simple, negative roots $r_n$ ($n:1,2,...,k$) and define $$Q_n=\lim_{x\to r_n}\frac{P(...
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How can I solve for $a$ in $\frac {k-\frac 32}a+\frac 1{2a^2}+\digamma(a)-\digamma(a+n)=0$, where $\digamma$ is the digamma function?

How can I solve for $a$ for this following equation? $$\frac {k-\frac 32}a+\frac 1{2a^2}+\digamma(a)-\digamma(a+n)=0,$$ where $\digamma$ represents the digamma function, i.e. it is defined as the ...
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fsolve and fimplicit for a non-linear equation

It's well known that from time to time nonlinear solver fsolve in Matlab gives controversial results but I'd like to ask and simultaneously share my experience of the exploitation of this command. I ...
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Find the least integer $k$ such that $\left[H_k\right]=n$ for a given positive integer $n$

At the moment I am curious about harmonic numbers at their properties and came across the following question. We are given $n\in\mathbb{N}$. We are to find the least integer $k$ such that $\left[H_k\...
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Growth of Digamma function

For $1\le \sigma \le 2$ and $t\ge 2$, $s=\sigma+it$ prove that $\displaystyle \frac{\Gamma'(s)}{\Gamma(s)}=O(\log t)$. From Stirling's formula we have, $\displaystyle \Gamma(s)\approx \sqrt{2\pi}\exp\...
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How to prove that $3^{x^2+x} (x+1)^{-x} \Gamma (x+1)\ge 1$ for $x>0$?

Let $$f(x)=3^{x^2+x} (x+1)^{-x} \Gamma (x+1).$$ Drawing a picture with any computer algebra system, it is obviously that $f(x) \ge 1$ on $[0,\infty)$. But How can we prove this? If we take ...
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Why is the digamma function written as ψ rather than ϝ or Ϝ?

The gamma function is written with the upper-case letter Γ, but the digamma function is usually written with the lower-case letter ψ. How did this notation come about? Why is it ψ rather than the ...
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Limit of sum of sequences at infinity

Given two sequences $$a_n=\int_0^1 (1-x^2)^n dx$$ and $$b_n=\int_0^1 (1-x^3)^n dx$$ ,($n\in N$) then find the value of $$L=\lim_{n\to \infty} (10 \sqrt [n]{a_n} +5\sqrt [n]{b_n})$$ My try: $$a_n=\...
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How do I solve for something inside a digamma function?

Say I have something like digamma(h(x)) -digamma(g(x))+f(x)=0 The digamma functions seem to prevent me from solving for x even if the functions f,g,h are very simple linear functions of x. Say ...
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Evaluation of $\sum_{n\geq1}\frac{1}{n}\ln(1+\frac{1}{n})$

Coming across the calculation of a special integral I get stuck on the following series, which I have given its integral representation : $$\text{J}=\sum_{n\geq1}\frac{1}{n}\ln\bigg(1+\frac{1}{n}\...
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Integration of digamma function

I was trying to perform the contour integral of the digamma function $\oint\limits_C \psi(z)\,dz$ on the neighborhood (a small circle $-k+re^{it}$, $k \in \mathbb{Z}$ ) of $k$, before actually ...
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Equation involving the psi function

I am trying to solve the following equation, and I can't simplify it even further. Is there any approximation or solution to this equation? $$\psi(r)+r\psi'(r)=\log r + 1$$ where $\psi$ is the Digamma ...
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A transformation formula for the digamma function

I derived an astonishing relationship that includes the digamma function:\begin{align}\upsilon(x,a)&=\gamma+\psi(1-aix)-\frac{a\pi}2\operatorname{csch}^2a\pi x+\frac{i\pi}{e^{2a\pi x}-1}-\frac1{x(...
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More elegant solution for showing that a distribution is unimodal

Consider the probability distribution for a discrete random variable $X$ with support $\left\{0,1,\ldots,50\right\}$: $$ \Pr\left[X=k\right] = \frac{{600\choose k}{400\choose 50-k}}{1000\choose 50} $$ ...
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About digamma function, does $\psi(a) =-\sum_{n=0}^{\infty} \frac{1} {n+a} ?$

I am trying to learn about digamma function and it's uses. For example series, I found somewhere this solution: $$\begin{align*} \sum_{n=0}^{\infty} \frac{1}{(3n+2)\left ( 3n+3 \right )} &= \sum_{...
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Explicit series for the minimum point of the Gamma function?

Is there any explicit series, product, integral, continued fraction or other kind of expression for the point at which $\Gamma(x)$ has a minimum in $(0,1)$? The decimal value can be found here http:/...
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How can I get the result?

I got a formula $$ - \frac{ \frac {\partial B(\lambda, \xi)}{\partial \lambda}}{B(\lambda, \xi)} = - \Psi(\lambda) + \Psi(\lambda + \xi) $$ Where $B$ is Beta function and $\Psi$ is digamma function. ...
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Integral - Combinations of logarithms, exponentials, and powers $\int_0^\infty x^{\nu-1}\ e^{-\mu x}\ \ln(x+a)\,dx$

$$\int_0^\infty x^{\nu-1}\ e^{-\mu x}\ \ln(x)\ dx = \frac1{\mu^\nu}\Gamma(\nu)\left[\psi(\nu)-\ln(\mu)\right] \quad\qquad [\Re\,\mu \gt 0, \quad \Re\,\nu\gt 0]$$ Hello, I found above equation on ...
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The meaning and definition of $\psi^{(-2)}(x)$, and the convergence of some related series involving the Möbius function

While I was playing with a CAS I find that makes sense the function $$\psi^{(-k)}(x),$$ for example $\psi^{(-2)}(x)$, where $\psi^{(n)}(x)$ denotes the $n$th derivative of the digamma function, see ...
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Prove that $-2\log(2) = -2 + \sum_{n=1}^{\infty}\frac{1}{n(2n+1)}$

How to prove that $$ -2\log(2) = -2 + \sum_{n=1}^{\infty}\frac{1}{n(2n+1)} $$ I know that this sum is equal to $\phi(1/2)+\gamma$ where $\phi(x)$ is the digamma function and $\gamma$ is the Euler-...
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Simple formula for $H_n = m + \alpha $?

Let $H_i$ be the $i$ th harmonic number. For a given positive integer $m$ we want to find the smallest possible positive integer value $n$ such that $H_n = m + \alpha $, where $\alpha > 0$. Let us ...
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Digamma Identities

I've been studying the digamma function lately and I often see it defined as the following: $$ \psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} $$ or $$ \psi(x) = \int_0^\infty \frac{e^{-t}}{t} - \frac{e^{-xt}...
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How I can got the partial sum of $\sum_{k=1}^{n}\frac{1}{(2k-1)}$?

It is clear that this sum $\sum_{k=1}^{n}\frac{1}{(2k-1)}$ is divergenet , but i don't succed to get it partial sum using standrad method ? Note: The sum is presented here in wolfram alpha by digamma ...
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What about $\int_0^1\frac{\log x}{\log(\gamma+\psi(x+1))}dx$, where $\gamma$ is Euler's constant and $\psi(x)$ the digamma function?

I was reading the list of integrals from Wikipedia, and I've created a variation of the integral $$\int \log_a(x)dx.$$ I show my example as a definite integral. I would like to know if is it possible ...
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Upper bound of $\left| \sum_{n=1}^N e^{2\pi i\psi(n+1)} \right|$, where $\psi(x)$ is the digamma function

Let $\psi(x)$ the digamma function, see its definition and relation with harmonic functions from this Wikipedia. Question. I am interested about if is known how to find an upper bound of $$ \left| \...
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Digamma function and asymptotic bounds

Knowing that $m\to \infty$, $n\to \infty$, $m^2<n^{2/3}$, I came across the following expression: $$\frac{2x^2}{\frac{x^2m^2}{n^{2/3}}+m^4n^{2/3}}$$ Playing a bit with said expression, I saw it ...
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Taylor series with logarithmic coefficients

The function $ \frac{-\gamma-ln(1-x)}{1-x} $ has series expansion: $$ \psi(1)+\psi(2)x+\psi(3)x^2+... $$ where $\psi(x)$ is the digamma function and $\gamma$ is the Euler-Mascheroni constant. Note ...
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Evaluate $\int_{0}^{\infty}e^{-x^2}\ln(x)dx$

Can a step-by-step answer be shown how to prove: $$\int_{0}^{\infty}e^{-x^2}\ln(x)dx = -\frac{{\pi^\frac{1}{2}}}{4}(\gamma+\ln(4))$$ I have a feeling differentiating under the integral sign could be ...
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Proof trigamma recurrence relation [closed]

How can I prove the digamma and trigamma recurrence relations, $$\psi(x+1)=\psi(x)+\frac{1}{x} \,?$$ I tried some advice for common recurrence relations, but it didn't open for me.
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Tricky partial rational sum

I'm looking for a simplification of $$ \sum _{k=r+1}^{2 r} \frac{2 k+2 r+1}{2 k^2-k (2 r+1)+2 r (r+1)}\:. $$ Mathematica gives a somewhat tautological result in terms of the digamma function $\psi$: $...
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Ratio of hypergeometric function with a “Harmonic twist” to a hypergeometric function

Continuing with a problem I am working which involves the work here, I am faced with the following expression. \begin{equation} \frac{1}{2\,_2F_1\left(\frac{1}{2},\frac{1}{2};1;z\right)} \sum_{n=0}^{\...
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Computing $\lim_{\varepsilon\to 0^{+}}\psi(\varepsilon)/\Gamma(\varepsilon)$ with asymptotic expansions

I have the following limit of which I want to compute: \begin{equation} \lim_{\varepsilon\to 0^{+}} \frac{\psi(\varepsilon)}{\Gamma(\varepsilon)}. \end{equation} For $\varepsilon\approx 0$ and $\...
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Partial sum formula for $\sum_{x=1}^n \frac{1}{bx - x^2}$ at specific limit without using digamma function

Given the following sum: $\sum_{x=1}^n \frac{1}{bx - x^2}$ (b is an integer constant) It appears to me that partial sum cannot be calculated without using Harmonic numbers, or Digamma function. ...
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Integrating $\psi'$ to get $\Gamma$

My question is about this proof here about Stirling's formula: A proof I found a while ago entirely relies on creative telescoping. Since $\frac{1}{n^2}-\frac{1}{n(n+1)}=\frac{1}{n^2(n+1)}$, ...
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Simultaneous transcendental equations involving digamma functions.

Is there a way (can I hope) to express the solutions of the following coupled transcendental equations in a closed form (in terms of Lambert, hypergeometric or other special functions) : \begin{...
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87 views

What about $\sum_{n=1}^\infty\frac{\mu(n)}{n}\psi\left(1+\frac{1}{n}\right)$ as $\frac{1}{2}$?

For integers $n\geq 1$ let $\mu(n)$ the Möbius function, and let $\psi(z)$ the Digamma function, see its definition and how is denoted and typeset in Wolfram Language, if you need it from this ...
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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}^{\infty }{\frac {dx}{x+...
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92 views

Limit involving gamma and digamma function

It can be shown that $\displaystyle \lim_{n \to \infty}\sqrt[n+1]{(n+1)!}- \sqrt[n]{n!}= \dfrac{1}{e}.$ If we let $f(x)= (\Gamma(1+x))^{\frac{1}{x}}$ and note that $f$ is differentiable over $(0, \...
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How to evaluate this integral without numerical methods?

$$ \int_{0}^{\pi/3}\frac{\mathrm{e}^x}{\cos\left(x\right)}\,\mathrm{d}x $$ I tried it with Trapezoidal and Simpson's got the correct answer which matches with Wolfram but how to find that form with ...
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371 views

What is the closed form of $\sum_{n\geq 1}(-1)^{n-1}\psi'(n)^2$?

This problem was proposed by Cornel Ioan Valean. What is the closed form of $$ S=\sum_{n\geq 1}(-1)^{n+1}\psi'(n)^2 $$ ? I recall that $\psi'(z)=\frac{d^2}{dz^2}\log\Gamma(z)=\sum_{m\geq 0}\...
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80 views

How fast does $\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right)$ go to $\alpha$?

In this question: $\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right)$ it is established that $$\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right) = \...