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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Proving $\int_0^1\frac{x^{b-1}}{1+x}dx=\psi(b)-\psi\left(\frac b2\right)-\log(2)$
Below is an integral that was sent to me along with the solution. Here is my attempt, I will show where exactly I am stuck:
$$\int_0^1\frac{x^{b-1}}{1+x}dx=\psi(b)-\psi\left(\frac b2\right)-\log(2)$$
...
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Equation relating the Riemann zeta function and the digamma function
While going through a few functional equations for the Riemann zeta function I stumbled upon one connecting it to the Digamma function $\psi(x)$, the formula is:
$$\zeta(s)=\frac{1}{s-1}+\frac{\sin(\...
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Integral derived from the correction factor of Sterling's approximation leading to a Puiseux series of polygammas
Sterling's approximation tells us that
$$x!\approx\sqrt{2\pi x}\left(\frac{x}{e}\right)^{x}$$
Taken and expanded upon from Spiegel, 1964, $\forall x>0\;\exists 0<\theta<1:$
$$\Gamma(x+1)=\...
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Rewriting a series of power-of-two Chebyshev polynomials with special functions or an integral (perhaps using Egorychev's method)
I would like to obtain an alternative expression (in terms of special functions or a more useful integral) for
$$f(x) = \frac{3}{7} + \frac{1}{2} \sum_{n = 0}^\infty 8^{-n} T_{2^n}\left(x\right) \\
= ...
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Zeros of $\Gamma’$.
I’ve seen in many places (e.g here https://arxiv.org/pdf/1409.2971) the following claims about the zeros of $\Gamma’$, without proof.
(1) It has only real zeros.
(2) The zeros are simple.
(3) Only one ...
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Generalization of digamma function
The digamma function has the following functional relation.
$$
\psi(z+1) = \psi(z) + \frac{1}{z}
$$
Can this be generalized? For $\alpha > 0$, I am interested to know if there is a family of ...
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Integral of principal value of $[\tanh(x+a)-\tanh(x+b)]/x$
How do you solve this integral involving the Cauchy principal value?
$$
\mathcal{P} \int_{-\infty}^{\infty} \frac{\tanh(x+a)-\tanh(x+b)}{x} dx \\
= \int_0^\infty \frac{\tanh(x+a)+\tanh(x-a)-\tanh(x+b)-...
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Derivative with respect to an index within a summation
First note that
$$ \sum_{n=0}^{\infty} \frac{\left(\frac{1}{2}\right)_{n}}{n!} \, x^n = \frac{1}{\sqrt{1-x}}, $$
where $(a)_{n}$ is the Pochhammer notation. The background is how to evaluate the ...
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Evaluating an infinite series with a function
There is an infinite series, I want to transform it into a function, with digamma functions or something else. I hope someone can provide some guidance and suggestions.
$$
E(x,y)=\sum_{n=-\infty}^{\...
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Where is the error in evaluating this series?$\sum_{n=1}\frac{1}{n^2+x^2}$
Where is the error in evaluating this series?
$$\sum_{n=1}\frac{1}{n^2+x^2}$$
My attempt:
$$\begin{align}\sum_{n=1} \frac{1}{n^2+x^2}&=\frac{1}{2ix}\sum_{n=1}\frac{1}{n-ix}-\frac{1}{n+ix}\\&=\...
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Function that grows asymptotically as $x\log(x)$ or $\log(x!)$, but is even? Function that grows as $\log(x)$ but is odd?
I'm trying to implement a model for predicting origin location - destination location traffic, from road traffic (if you are interested in the model, it is from this paper: Bell1983). The author ...
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Is there other method to get a nicer answer for the integral $\int_0^{\infty} \frac{\left(\sqrt x-1\right)^2}{\left(x^2+1\right) \ln^2 x} d x $?
Recently, I was requested to evaluate an integral
$$\int_0^{\infty} \frac{\left(\sqrt x-1\right)^2}{\left(x^2+1\right) \ln ^2x} d x.$$
I then try to use the Feynman’s trick by considering the ...
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Alternative method of evaluating $\int_0^{\frac{\pi}{2}} \sin ^{2 n} x \ln (\tan x) d x $?
LATEST EDITION
Glad to share with you that we had found below as an answer, in general, that
$$\boxed{ \int_0^{\frac\pi2} {\sin^n x} \ln{(\tan x)} \,dx =\frac{\sqrt{\pi}}{4 \Gamma\left(\frac{n}{2}+1\...
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approximation to digamma function
I was learning about the harmonic series back in college to which the professor said "There is no known closed form for the harmonic sum", I felt that was strange given that the sum didn't ...
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Sum of over primes $p$ of $x^{-p}$
I was playing around with the following series
$$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(\...
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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 ...
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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:
$$I=2\int_{0}^{\infty}\frac{\sin(x)}{e^...
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Does $\sum_{r = 1}^n \ln\left(\frac{1 + r}{r}\right) = \ln (\Gamma(n + 2)) - \ln (\Gamma(n+1))$? If so, why?
When attempting the evaluate the integral $\int_0^1 \{\ln(x)\}$, where $\{ x \}$ is the fractional part function, I came across the following sum:
$$\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}\...
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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 ...
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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 ...
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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
$$
ψ\left(\frac{p}{q}\right) = -\gamma - \frac{\pi}{2}\cot\left(\pi\frac{p}{q}\right) - \ln(q) + \frac{1}{2}\sum_{k=1}^{q-1} \cos\left(...
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What will be this limit? I'm having a hard time finding this....
What will be this limit?
\begin{equation}
\lim_{\ n\to\infty}\ \left(\frac{\left(n!\right)}{n}\right)^{\frac{1}{n}}=P\
\end{equation}
I tried it like this:
its in the form (infinity)^(0), so taking ...
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Seeking for other methods to evaluate $\int_0^{\infty} \frac{\ln \left(x^n+1\right)}{x^n+1} dx$ for $n\geq 2$.
Inspired by my post, I go further to investigate the general integral and find a formula for
$$
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 ...
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