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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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### 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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### 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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### 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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### 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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### 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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