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.

168 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
98 views

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 ...
64 views

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 ...
206 views

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, ...
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 (...
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 ...
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 ...
38 views

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 \...
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 ...
44 views

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