gamma function – algebraic manipulation on equation

An equation involving the poisson point process is formulated as:

$prod_{j=1}^{K}exp(-2pilambda_j(sp_j)^{2/alpha}int_0^{infty}rint_0^{infty}e^{-t(1+r^{alpha})}dtdr)$.

Some algebraic manipulations are carried out and the equation is rewritten as:

$exp(-s^{2/alpha}C(alpha)sum_{i=1}^{K}lambda_iP_i^{2/alpha})$, where $C(alpha)=frac{2pi csc(frac{2}{alpha})}{alpha}$.

I only know the manipulation is related to Gamma function. I want to know the datails about the manipulation.

Thanks a lot! : )

pr.probability – Inaccurate results for the analytical expression of $mathbb{E}left[ a mathcal{Q} left( sqrt{b } gamma right) right]$

I’m trying to plot a graph for the following expectation

$$mathbb{E}left( a mathcal{Q} left( sqrt{b } gamma right) right)=a 2^{-frac{kappa }{2}-1} b^{-frac{kappa }{2}} theta ^{-kappa } left(frac{, _2F_2left(frac{kappa }{2}+frac{1}{2},frac{kappa }{2};frac{1}{2},frac{kappa }{2}+1;frac{1}{2 b theta ^2}right)}{Gamma left(frac{kappa }{2}+1right)}-frac{kappa , _2F_2left(frac{kappa }{2}+frac{1}{2},frac{kappa }{2}+1;frac{3}{2},frac{kappa }{2}+frac{3}{2};frac{1}{2 b theta ^2}right)}{sqrt{2} sqrt{b} theta Gamma left(frac{kappa +3}{2}right)}right)$$
where $a$ and $b$ are constant values, $mathcal{Q}$ is the Gaussian Q-function, which is defined as $mathcal{Q}(x) = frac{1}{sqrt{2 pi}}int_{x}^{infty} e^{-u^2/2}du$ and $gamma$ is a random variable with Gamma distribition, i.e., $f_{gamma}(y) sim frac{1}{Gamma(kappa)theta^{kappa}} y^{kappa-1} e^{-y/theta} $ with $kappa > 0$ and $theta > 0$.

This equation was also found with Mathematica, so it seems to be correct.

Follows some examples, where I have checked the analytical results against the simulated ones.

When $kappa = 12.85$, $theta = 0.533397$, $a=3$ and $b = 1/5$ it returns the correct value $0.0218116$.

When $kappa = 12.85$, $theta = 0.475391$, $a=3$ and $b = 1/5$ it returns the correct value $0.0408816$.

When $kappa = 12.85$, $theta = 0.423692$, $a=3$ and $b = 1/5$ it returns the value $-1.49831$, which is negative. However, the correct result should be a value around $0.0585$.

When $kappa = 12.85$, $theta = 0.336551$, $a=3$ and $b = 1/5$ it returns the value $630902$. However, the correct result should be a value around $0.1277$.

Therefore, the issue happens as $theta$ decreases. For values of $theta > 0.423692$ the analytical matches the simulated results. The issue only happens when $theta <= 0.423692$.

I’d like to know if that is an accuracy issue or if I’m missing something here and if there is a way to correctly plot a graph that matches the simulation.

calculus and analysis – Integral of $r frac{2^{r-1} log (2) e^{-frac{sqrt{2^r-1}}{b}} left(2^r-1right)^{frac{d}{2}-1}}{b^d Gamma (d)}$ with Mathematica

I’m trying to find the integral given below with Mathematica

$int_0^{infty } r frac{2^{r-1} log (2) e^{-frac{sqrt{2^r-1}}{b}} left(2^r-1right)^{frac{d}{2}-1}}{b^d Gamma (d)} , dr$

However, it takes too long for it to return something and when it returns it outputs the same integral.

$int_{0}^{infty } frac{2^{r-1} r log (2) b^{-d} e^{-frac{sqrt{2^r-1}}{b}} left(2^r-1right)^{frac{d}{2}-1}}{Gamma (d)} , dr$

I’d like to figure out the solution for this integral.

coding theory – Variance Gamma parameter estimation in R studio

I am using the vgFit function in R Studio to estimate parameters for stock prices. I converted the stock prices to a vector and I get this error. My code was:

param<-c(0,1,0,1) 
fit<-vgFit(c(x), param=param)

The error is:

Error in optim(paramStart, llsklp, NULL, method = startMethodSL, hessian = FALSE,  : 
  function cannot be evaluated at initial parameters

analytic number theory – Is every modular function on $Gamma$ univalent?

Suppose $f$ is a modular function on $Gamma$ then it has a fundamental region $R_L$. Since $f$ is modular, it can be expressed as a rational function of $J(tau)=frac{g_2^3(tau)}{Delta(tau)}$. To prove this we actually construct a rational function
$$g(tau)=frac{prod(J(a_i)-J(tau))}{prod(J(b_i)-J(tau))}$$
where the $a_i,b_i$ are zeros and poles of $f$. It seems to me that this construction implies the possibility of having multiple poles in the fundamental region. I know for sure that $J$ has only one pole since it attains every value exactly once.


Since a modular function is univalent if and only if its fundamental region has genus $1$ and if we identify the edges of $R_Gamma$ it would become homeomorphic to $mathbb{P}^1$, any modular function that has the fundamental region $R_Gamma$ should be univalent, that is, any modular function is univalent in the closure of its fundamental region.

This really confused me and I am not sure if this is correct.

Fitting data to left skewed gamma distribution

How can I fit the following two set of data to a left skewed gamma function, which I what I think should fit the data best?:

data 1 is here: https://pastebin.com/X2HTjTP7
data 2 is here: https://pastebin.com/8Rh4BHDT

Is there any other suggestion of what would be the best distribution or equation to fit the data?

A picture of how data 1 looks is here:

data 1

A picture of how data 2 looks is here:

data 2

Thank you in advanced,

How to calculate gamma function values ​​for quaternions with Mathematica?

How can we calculate the values ​​for that? gamma Function with Quaternions on Mathematica?

For example:

Gamma(5) = 24

N(Gamma(I)) = -0.15495 - 0.498016 I

N(Gamma(1 + 2 I)) = 0.151904 + 0.0198049 I

N(Gamma(Quaternion(5, 0, 0, 0))) = Gamma(Quaternion(5., 0., 0., 0.)) (* which is wrong *)

How can we calculate specifically?

Gamma(Quaternion(1, 2, 3, 4))?

Tries:

N(Gamma(FromQuaternion(Quaternion(1, 2, 0, 0)))) (* works *)
N(Gamma(FromQuaternion(Quaternion(1, 2, 3, 4)))) (* doesn't work *)


QuaternionPower(x_, y_) := E^(y ** Log(x))
QuaternionGamma(z_) := Integrate(QuaternionPower(E, -x + Log(x)*(z - 1)), {x, 0, Infinity})
N(QuaternionGamma(FromQuaternion(Quaternion(1, 2, 0, 0)))) = 0.151904 + 0.0198049 I (* works *)
N(QuaternionGamma(FromQuaternion(Quaternion(1, 2, 3, 0)))) = ConditionalExpression(Gamma((1.  + 2. I) + 3. J), Re(J) > -0.333333) (* doesn't work *)

Logic – If a set of sentences $ Gamma $ has an infinite model, does it also have a denumerable model?

I think that's wrong, since we can only conclude from the Downward-Lowenheim-Skolem theorem $ Gamma $ has an enumeration model, it is not shown whether this enumeration model is infinite. Certainly ours $ Gamma $ already has an infinite model, but nothing says that its enumerable model must be identical to its infinite model.

I can't see how this claim is true, but I think it should be true?