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.

Filter by
Sorted by
Tagged with
0
votes
1answer
33 views

Characterizing $\frac{m^k - e^{-x}(m+1)^k}{k!}$

I'm trying to characterize the behavior this function: $l_k(m,x)=\frac{m^k - e^{-x}(m+1)^k}{k!}$. I was wondering whether either of these functions are well-known in the probability theory and ...
4
votes
2answers
117 views

Alternative approaches to showing that $\Gamma'(1/2)=-\sqrt\pi\left(\gamma+\log(4)\right)$

Starting from the definition of the Gamma function as expressed by $$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x}\,dx\tag1$$ we can show that the derivative of $\Gamma(z)$ evaluated at $z=1/2$ is given by $$\...
1
vote
2answers
52 views

The antiderivative of $\sum_{n\gt 0}\frac{x}{n(x+n)}$

I tried to calculate $\int\sum_{n\gt 0}\frac{x}{n(x+n)}\, \mathrm dx$: $$\begin{align}\int\sum_{n\gt 0}\frac{x}{n(x+n)}&=\sum_{n\gt 0}\frac{1}{n}\int\left(1-\frac{n}{x+n}\right)\, \mathrm dx \\&...
2
votes
1answer
63 views

On a log-gamma definite integral

A very famous log-gamma integral due to Raabe is $$\int_0^1 \log \Gamma (x) \, dx = \frac{1}{2} \log (2\pi).$$ Several proofs of this result can be found here. I would like to known about the ...
1
vote
1answer
31 views

Understanding this property of the digamma function: $\pi(x + \frac{1}{2}) = -\gamma - 2\ln 2 + \sum\limits_{k=1}^n\frac{2}{2k-1}$

I am reading through the Wikipedia article on the digamma function where the digamma function is compared to the Harmonic Numbers defined as $H_n = \sum\limits_{k=1}^{n} \dfrac{1}{k}$. The first ...
1
vote
2answers
80 views

Is this a correct way to use the digamma to analyze whether a ratio of gamma functions is increasing?

Let: $\pi(x)$ be the prime counting function $\psi(x)$ be the digamma function Given that $\pi(x) < \dfrac{1.25506n}{\ln n}$ (see here), let: $$f(n) = \frac{\Gamma\left(2n + 1 - \frac{1.25506n}{\...
1
vote
1answer
52 views

Is the following ratio of gamma functions increasing: $\frac{\Gamma(2n - \frac{1.25506n}{\ln n})}{\Gamma(n)^2}$?

For $n > 1$, is the following ratio of gamma functions increasing: $\dfrac{\Gamma(2n - \frac{1.25506n}{\ln n})}{\Gamma(n)^2}$ I suspect that it is at some point where $n > 1$. I would like ...
0
votes
2answers
40 views

Why does $\lim_{k \to \infty}\psi(k)$ diverge?

This question is motivated by the fact that $$\psi(k)+\gamma=H_{k-1}$$ This, of course, implies that $$\lim_{k \to \infty}\psi(k)+\gamma = \lim_{k \to \infty}H_{k-1}$$ Since $H_k$ diverges, it ...
0
votes
0answers
23 views

Is there any possibility that there exists a formula for the partial sums of the harmonic series? If not, why? [duplicate]

I have been researching the harmonic series, $H_k$ and the Euler-Mascheroni constant, $\gamma$, quite a bit recently. I understand that the diagamma function has this property: $$\psi^{0}(k+1) + \...
0
votes
0answers
42 views

Sum involving 2nd antiderivative of the digamma function.

Help evaluate the following sum: $$\sum_{n=1}^{\infty}\left(\frac{1}{4n-1}-\frac{1}{4n-1}\ln\left(2n-1\right)+\frac{4}{\left(4n-1\right)\left(4n-3\right)}\left(\psi^{(-2)}\left(2n-\frac{1}{2}\right)-\...
0
votes
0answers
32 views

An inequality for the Gamma function

Let $0<x<1$ and $0<y<1$ then we have : $$\Gamma{(x)}\Gamma(y)\leq \Gamma\Big(\frac{x+y}{2}\Big)\Gamma\Big(\frac{(x)\psi^{(0)}(x)+(y)\psi^{(0)}(y)}{\psi^{(0)}(x)+\psi^{(0)}(y)}\Big)$$ ...
0
votes
0answers
35 views

Closed-form expression for an integral involving the Digamma function

I have the following integral for which I'm trying to find a closed-form: $\int^\infty_0 \left[2 X^t \log(X) - X\left\{\log(2) - 2 \log({\theta}) + \psi\left(1+\frac{t}{2}\right)\right\}\right]^2 t \...
3
votes
1answer
83 views

Does $\sum_{n=1}^{\infty} (-1)^n \zeta(n + 1)$ converge to $-1$?

According to my calculations $$\sum_{n=1}^{\infty} (-1)^n \zeta(n + 1) = -1$$ but for example WolframAlpha says that the sum doesn't converge, so have I done anything wrong (my calculations down below)...
0
votes
0answers
17 views

Is the derivative of gammaln((x+1)/2) equal 0.5psi((x+1)/2)?

I just want to know if the differentiation of $\ln\left(\Gamma(\cdots)\right)$ follows the usual rules. So is the following true? $$\dfrac{\mathrm d}{\mathrm dx}\ln\left(\Gamma\left(\dfrac{x+1}2\...
5
votes
2answers
213 views

Evaluation of $\int\limits ^{\infty }_{0}\frac{x}{\left( x^{2} +1\right)^2\left( e^{tx} +1\right)} dx$

I want to show that $$\int\limits ^{\infty }_{0}\frac{x}{\left( x^{2} +1\right)^2\left( e^{tx} +1\right)} dx=\frac{\psi^{(1)}(\frac{t}{2\pi})-4\psi^{(1)}(\frac{t}{\pi})}{8\pi}t+\frac14 $$ where $\...
5
votes
1answer
91 views

Is there a closed form for the polygamma function?

Mathematica gave me that $$ \sum_{k=n}^\infty \frac1{k^2} = \texttt{PolyGamma[1,n]}. $$ However, in all my attempts to simplify and approximate the number as a decimal, it kept leaving it in terms of ...
4
votes
2answers
176 views

Find Harmonic Numbers for Imaginary and Complex Values

The Normal definition of Harmonic numbers with $ n \in \mathbb{N} $ is $$ H_n = \sum_{k=1}^{n}\frac{1}{k} \tag{1}\label{eq1A} $$ This can be expanded to $ n \in \mathbb{C} $ $$ H_n = \psi_0(n+1) + ...
1
vote
1answer
55 views

How one can show that the gamma function is a strictly increasing function on the interval $(1.4616,+∞)$.

How one can show that the gamma function is a strictly increasing function on the interval $(1.4616,+∞)$.
1
vote
1answer
96 views

Gauss's proof that the Digamma function equals $\int_0^{\infty}(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1-e^{-t}})dt$.

I was reading about the Digamma function, defined as: $$\psi(z) = \frac{d}{dx}\ln( \Gamma(z)) = \frac{\Gamma ' (z)}{\Gamma(z)}$$ And the following integral representation of $\psi(z)$ was given for $z:...
1
vote
1answer
35 views

ratio of gamma function

How can we show that $f(x)=\Gamma(x+s)/\Gamma(x)$ is an increasing function for $x>1$ and $0<s<1$. I have checked by plotting in sage or wolframalpha. So the result is true for sure. I ...
5
votes
2answers
173 views

Compute $\lim_{x\rightarrow 0 }\biggr ( \dfrac{1}{x}\ln (x!)\biggr )$

I want to compute the following limit $$\lim_{x\rightarrow 0 }\biggr ( \dfrac{1}{x}\ln (x!)\biggr )$$ Since factorial is only defined for integers, we must use the gamma function. $$\lim_{x\...
2
votes
2answers
95 views

Evaluation of a digamma series involving golden-ratio

Let $\varphi =\frac{1}{2} \left(\sqrt{5}+1\right), a=\tan \left(\frac{\sqrt{5} \pi }{2}\right)$, then I was given the following identity: $$\sum _{n=1}^{\infty } \frac{\psi ^{(0)}(n+\varphi)-\psi ^{(0)...
1
vote
1answer
178 views

Real and imaginary parts of $\ln \Gamma(i b)$

Can one obtain the real and imaginary parts of $\ln \Gamma (i b)$ in terms of simpler functions? ($b$ is a positive number.)
0
votes
3answers
100 views

Is there a closed form for series of $\sum_k \frac{x^k}{1-x^{2k+1}}$

is there a way to evaluate $\sum_{k=0}^{+\infty} \frac{x^k}{1-x^{2k+1}}$ in terms of popular functions or even in terms of the q-digamma function? $0<x<1$ I tried to write the denominator as ...
0
votes
2answers
60 views

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 < 0. Plugging it into Wolfram Alpha ...
3
votes
2answers
65 views

Prove $\lim_{n \to \infty} ( \frac{\psi(-1/2+n i )- \psi(-1/2-n i)}{2i}- \frac{n}{n^2+1/4} ) = \frac{\pi}{2}$

Consider the sequence: $$A_n= \frac{\psi(-1/2+n i )- \psi(-1/2-n i)}{2i}- \frac{n}{n^2+1/4} $$ How would you prove that: $$\lim_{n \to \infty} A_n= \frac{\pi}{2}$$ This sequence converges ...
3
votes
0answers
105 views

Prove a new digamma finite sum

There are a few interesting finite sums of digamma of a rational argument listed on Wikipedia (from this paper). One of them is the following: $$\sum _{m=1}^{N-1}\psi \left({\frac {m}{N}}\right)\cdot ...
3
votes
2answers
92 views

Prove a tough limit involving the digamma function

Here I have a limit to which I arrived while working on a seperate integral through Mellin Transforms. $$\lim\limits_{s\to -1^{-}}\Big[\psi_{(0)}(s)-\frac{\pi}{2}\tan\left(\frac{\pi s}{2}\right)\Big]...
2
votes
1answer
111 views

psi digamma function

Is well known that $$\psi(x)-\psi(-x)=-\pi \cot(\pi x) - \frac{1}{x}.$$ I am wondering if a similar property holds for the following function, $$D_{\beta,\gamma}(x) = \psi(\beta x)-\psi(-\gamma x),\ \...
9
votes
3answers
274 views

Help with $-\int_0^1 \ln(1+x)\ln(1-x)dx$

I have been attempting to evaluate this integral and by using wolfram alpha I know that the value is$$I=-\int_0^1 \ln(1+x)\ln(1-x)dx=\frac{\pi^2}{6}+2\ln(2)-\ln^2(2)-2$$ My Attempt: I start off by ...
1
vote
1answer
64 views

How to solve $\int_0^\infty \frac{\pi^2 e^{-x}}{6x}-\frac{ \operatorname{Li}_2(e^{-x})}{1-e^{-x}}dx$?

While investigating the sum $$S=\sum_{n=1}^\infty \frac{\psi_0(n)} {n^2}$$ I used the integral representation of the Digamma function to get$$S=\int_0^\infty \frac{\pi^2 e^{-x}}{6x}-\frac{ \...
0
votes
0answers
130 views

Find limit of sequence defined by sum of previous terms and harmonics

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function? I tried to ...
3
votes
2answers
120 views

Show that : $\sum_{k=1}^{\infty}\frac{i^{k(5k+1)}}{k(k+1)}=1-\frac{π}{2}$

Show that $S=\displaystyle\sum_{k=1}^{\infty}\frac{i^{k(5k+1)}}{k(k+1)}=1-\frac{π}{2}$ My try : $S=\displaystyle\sum_{k=1}^{\infty}\frac{e^{iπk(5k+1)/2}}{k(k+1)}$ $=\displaystyle\sum_{k=1}^{\...
1
vote
1answer
48 views

Uniform convergence of digamma function

Let $F_n$ be a real valued function, $$F_n(r) = \frac{r}{n}(\psi(n) - \psi(r)$$ where $\psi$ is a digamma function. Let the sequence of functions $\{g_n\}$ be defined by $g_n(x) := F_n(nx)$. Show ...
0
votes
1answer
105 views

Compute in closed form that $S=\sum_{n=1}^{\infty}\frac{1}{6n^5+15n^4+10n^3-n}$

Compute the following sum : S=$\sum_{n=1}^{\infty}\frac{1}{6n^5+15n^4+10n^3-n}$ My attempt : Use partial fraction : $6n^5+15n^4+10n^3-n=n(n+1)(2n+1)(3n^2+3n-1)$ $S=\sum_{n=1}^{\infty}(\frac{9(2n+...
1
vote
1answer
71 views

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 ...
0
votes
1answer
78 views

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) ...
1
vote
1answer
184 views

digamma function inverse and special value

What the inverse of the digamma function?, and specifically how can I write the x for:$$ ψ(x)=1$$$[x ≈ 3.20317146837693106929448152]$ without the digamma?
0
votes
1answer
277 views

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 ...
2
votes
1answer
106 views

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\...
0
votes
2answers
143 views

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 ...
0
votes
1answer
47 views

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 ...
1
vote
1answer
118 views

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.
2
votes
1answer
65 views

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(...
0
votes
0answers
57 views

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 ...
0
votes
0answers
55 views

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 ...
6
votes
2answers
189 views

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\...
2
votes
1answer
49 views

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 ...
4
votes
0answers
49 views

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 ...
3
votes
2answers
173 views

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=\...