1
vote
1answer
58 views

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 ...
0
votes
1answer
36 views

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 $\Gamma(2k+2)$?
1
vote
1answer
103 views

How to evaluate $\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$

How to evaluate $$\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$$, where n is integer > 0? I know the gamma function formula will give $$ ...
6
votes
1answer
108 views

Proof of $\Gamma(z) e^{i \pi z/2} = \int_0^\infty t^{z-1} e^{it}\, dt$

I am trying to prove the identity $$\Gamma(z) e^{i \pi z/2} = \int_0^\infty t^{z-1} e^{it}\, dt$$ for $0 < \Re(z) < 1$, starting from the integral definition of the gamma function $$\Gamma(z) = ...
2
votes
1answer
158 views

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. ...
4
votes
1answer
286 views

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 ...
0
votes
0answers
210 views

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: ...
4
votes
2answers
77 views

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 $$2\frac{\mathrm dI(a)}{\mathrm da}+aI(a)=0$$ ...
2
votes
0answers
59 views

gamma funtion and estimates-typo or mistake?

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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
0answers
139 views

Determining well-definedness for functions

How does one determine well-definedness in analytical continuation for $\Gamma(s)\zeta(s)$ function? Firstly: $$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$ ...
6
votes
1answer
1k views

Some questions about the gamma function

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) = ...