# Tagged Questions

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### Is there a nice solution to the equation $\Psi(x)=\ln(\pi)$ with a positive real $x$?

I tried to find a nice solution to the following equation: $$\Psi(x)=\ln(\pi)$$ with $x\in\Bbb R_{\ge0}$ and where $\Psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$. Is there a nice expression for x satisfying ...
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### Question about gamma function recurrence formula.

I know that $\Gamma(k+1)=k\Gamma(k)$. But I am not sure about $2k\,\Gamma(2k)=$? Can anyone help me out with this? Is it $\Gamma(2k+1)$ or $\Gamma(2k+2)$?
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### differentiate log Gamma function

I am working with the log of negative binomial distribution $NegBin(r,p)$. I need to differentiate the following with respect to $r$ such that at the end, I am NOT left with $r$ in factorial form. ...
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### Summation of Gamma functions and fractions

Can anyone explain to me what identities are used to get the solution: $$\sum_{t=1}^{n} \frac{\Gamma(t+a)}{\Gamma{(t)}}=\frac{n\Gamma(a+n+1)}{(a+1)\Gamma{(n+1)}}$$ This is only the last step of a ...
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### bayesian estimation on the poisson distribution

Suppose X~Poisson($\lambda$) and $\lambda$~Gamma($\alpha,\beta$). Find the posterior distribution and the Bayesian estimator of $\lambda$. Thus the prior distribution is: ...
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### Solution to the ODE $\qquad2\frac{\mathrm dI(a)}{\mathrm da}+aI(a)=0$

I am working on this question Show that the function $$I(a)=\int\limits_0^\infty e^{-u^2}\cos(au)\,\mathrm du$$ satisfies the differential equation $$2\frac{\mathrm dI(a)}{\mathrm da}+aI(a)=0$$ ...
In one of the lecture notes I've found that $C_n$ $$C_n= \begin{cases} \frac{n!}{\sqrt 2 \Gamma((n/2+1)}\pi^{-1/42^{-n/2}(n!)^{-1/2}} & n\text{ even} \\[4mm] \frac{2(n!)}{(\sqrt2n+1/(\sqrt2 ... 1answer 4k views ### Integral of x^2 \exp(-ax^2) using the gamma function? I'm struggling with this integral:$$\int_{-\infty}^{\infty}x^{2}e^{-ax^{2}}dx = \frac{\sqrt{\pi}}{2a^{3/2}}$$Is there a way to do this integration without using integration by parts and then ... 2answers 3k views ### Moment generating function of Gamma distribution I'm trying to show that as \alpha tends to 0, the gamma distribution$$\Gamma(\lambda,\alpha),$$is properly standardised, tends to the standard normal distribution. I have figured out that the ... 0answers 139 views ### Determining well-definedness for functions How does one determine well-definedness in analytical continuation for \Gamma(s)\zeta(s) function? Firstly:$$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1 ...
Show that $\Gamma(y) = \int_0^{\infty}{e^{-x}x^{y-1}\,dx}$ is finite for $y>0$ both as an improper Riemann integral and as a Lebesgue integral. Show \$\Gamma'(y) = ...