Tag: gamma

sequences and series – Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?)

This question arose from Amdeberhan’s question, the evaluation of a double integral, which can be reduced to the evaluation of this series:
$$sum _{n=0}^{infty } frac{Gamma left(n+frac{1}{2}right)^2 Gamma left(n+frac{s}{2}right)}{Gamma (n+1)^2 Gamma (n+s)}=frac{pi ^2 2^{1-s} Gamma left(frac{s}{2}right)}{left(Gamma left(frac{3}{4}right) Gamma left(frac{s}{2} +frac{1}{4}right)right)^2},;;{rm Re},s>0.$$
The evaluation of the sum is Mathematica output. Can someone enlighten me as to how this calculation proceeds?

I went so far as to pay for Wolfram Alpha Pro, hoping that it would disclose the steps, but to no avail. What is even more frustrating is that for $s=1$ the right-hand-side is the square of a complete elliptic integral, which is also recognized immediately by Mathematica and was the original question in the cited post, so far without a conclusive answer.

Gamma Correction and Shadow Detail bit allocation

I’m having slight difficulties understanding how a gamma correction increases details in the shadows(where our eyes are more sensitive). Once the bits have been been reallocated to the shadows after applying an INVERSE GAMMA/GAMMA CORRECTION in the camera wouldn’t all that detail just be lost as the monitor would apply a GAMMA to counter the inverse gamma thus bringing the image luminance back to a linear function. Or are the code values saved after gamma correction and only the brightness is brought down.

I’ll use an example I took from the video “Diving into dynamic range” from Filmmaker IQ on youtube.

From what I understood, in his example he uses an 8 STOP(the triangles represent the stops) camera with an 8 bit depth(Not exactly sure about this I got abit confused here. Please correct me If I’m wrong)

enter image description here

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Basically once the GAMMA of the screen is applied to the above OETF GAMMA curve and it goes back to the LINEAR one why wouldn’t we loose all the details in the shadow again??

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Here’s the link to the video just in case: https://www.youtube.com/watch?v=2sshGdMgJxQ

nt.number theory – Computing the $p$-adic gamma function $Gamma_p$

Let $p>2$ be a prime. For $n in mathbb{Z}^+$ we can define
begin{equation}
F(n) = (-1)^n prod_{1<i<n, pnmid i} i.
end{equation}
Since $mathbb{Z}$ is dense in $mathbb{Z}_p$, we can extend $F$ uniquely to a continuous function on $mathbb{Z}_p$, which is the $p$-adic gamma function $Gamma_p$.

Is there any good way to explicitly compute $Gamma_p(a/b)$ where $a,b in mathbb{Z}$ such that $a/b in mathbb{Z}_p^{times}$? Specifically, I’m looking at $Gamma_5(4/11)$.

A bad idea is to evaluate $F(a_n)$, where $a_n$ are the partial sums of the $5$-adic expansion of $4/11$. I don’t think this would work since $F(a_n)$ becomes too difficult to compute the further you go in the expansion.

I’m aware that some values can be computed via the Gross-Koblitz formula, but I don’t know how or if I can apply it to my situation.

Any help is appreciated!

ct.category theory – $Gamma: mathcal C to text{Fun}(mathcal Z, mathcal C)$ has a left adjoint iff $F in text{Fun}(mathcal Z, mathcal C)$ has a colimit

Let $mathcal Z$ be a small category and $mathcal C$ any category. We then consider the category $text{Fun}(mathcal Z, mathcal C)$ the category of functors from $mathcal Z$ to $mathcal C$ and the functor $Gamma: mathcal C to text{Fun}(mathcal Z, mathcal C)$ such that
$$
Gamma(C):
begin{pmatrix}
mathcal Z &longrightarrow &mathcal C\
Z & longmapsto & C\
f & longmapsto & 1_C
end{pmatrix}
$$
where $f: Z to Z’$ and $1_C$ is the identity morphism on $C$.

I am trying to show that $Gamma$ has a left adjoint if and only if the colimit of every functor $F: mathcal Z to mathcal C$. The “if” part is not too hard but I have some difficulties to show the “only if” part.

Here is my idea: Suppose that there is $Omega : text{Fun}(mathcal Z, mathcal C) to mathcal C$ such that
$$theta_{F, C}: text{Hom}_mathcal C(Omega(F), C)cong text{Nat}(F, Gamma(C)).$$
For each $alpha in text{Nat}(F, Gamma(C))$ we can associate a cocone, i.e. for $f: Z to Z’$ the following diagram commutes
$$
begin{array}{ccc} F(Z)&xrightarrow{F(f)} &F(Z’)\ searrow&&swarrow\
&Gamma(C)(Z) = C = Gamma(C)(Z’)& end{array}
$$
where the arrows $F(Z) to C$ and $F(Z’) to C$ are given by $alpha_Z$ and $alpha_{Z’}$ respectively. Because of the above isomorphism, we can associate to $alpha$ a unique morphism $theta^{-1}_{F, C}(alpha) =u: Omega(F) to C$ so $Omega$ is a good candidate to be the colimit of $F$. We just have to find a family of morphism $mu_Z: F(Z) to Omega(F)$ such that $alpha_Z = u circ mu_Z$. My guess is that this family of morphism is given by the unit of the adjunction $eta_F in text{Nat}(F, Gamma(Omega(F)))$ but I am not able to show that
$$alpha_Z = u circ (eta_F)_Z = theta^{-1}_{F, C}(alpha) circ (theta_{F, Omega(F)}(1_{Omega(F)}))_Z$$
where the last equality comes from the expression of $eta_F$ in term of $theta_{F, Omega(F)}$. Does it seem right ? Does anyone know how to conclude ?

probability distributions – Posterior for Gamma prior and Gamma likelihood with known shape

Following “Introduction to Mathematical Statistics” of Hogg et al. (Exercise 11.2.2) I am trying to calculate the posterior distribution of $theta$ with the following information given:
Let $X_1,X_2,…,X_{10}$ be a random sample of size $n=10$ from a gamma distribution with $alpha = 3$ and $beta=frac{1}{theta}$. Suppose we believe that $theta$ has a gamma distribution with $a=10$ and $b=2$.
I tried to calculate the posterior in general form first and then substitute the variables with the concrete values above. For the Prior I have:
$$ f_{Theta}(theta)=frac{b^a}{Gamma(a)} cdot theta^{a-1} cdot e^{-btheta} $$
And for the likelihood:
$$ f_{Xmid Theta}(xmidtheta)=frac{beta^alpha}{Gamma(alpha)} cdot x^{alpha-1} cdot e^{-beta x}$$
Therefore for the posterior the following should be true:
$$ f_{Theta mid X_1,X_2,…,X_n}(thetamid x_1,x_2,…,x_n)propto prod_{I=1}^{n}frac{beta^alpha}{Gamma(alpha)} cdot x_i^{alpha-1} cdot e^{-beta x_i} cdot frac{b^a}{Gamma(a)} cdot theta^{a-1} cdot e^{-btheta}$$
Now removing all terms independent of $theta$ my result is the following:
$$f_{Theta mid X_1,X_2,…,X_n}(thetamid x_1,x_2,…,x_n)propto beta^{nalpha} cdot e^{-betasum_{I=1}^{n}x_i}cdottheta^{a-1}cdot e^{-btheta}$$
Inserting $beta=frac{1}{theta}$ and aggregating:
$$f_{Theta mid X_1,X_2,…,X_n}(thetamid x_1,x_2,…,x_n)propto theta^{a-nalpha-1} cdot e^{-frac{sum_{I=1}^{n}x_i}{theta}-btheta}$$
Finally I am struggling to relate this to any common distribution. Also taking the following hint from the exercise into account: “Can the posterior distribution be related to a chi-square distribution?”
Can anybody help?

ct.category theory – Functor category with only isomorphisms in the Gamma space construction

I have not read Marcolli’s paper, but it sounds like the construction you are describing goes back to Segal in “Categories and cohomology theories”.

I think the short answer to your question is people take the category of isomorphisms because in this way one gets the most interesting homotopy type. Suppose for example you took the category of summing functors and all natural transformations between them. Since the category $C$ has has a zero object, it follows that the category of summing functors has an initial object, so the geometric realization of the category of summing functors would be contractible.

Segal’s motivation was to construct interesting infinite loop spaces out of categories. The idea (or rather my crude simplification of it!) is that if $C$ is a symmetric monoidal category, then its classifying space $BC$ is not quite an infinite loop space, but it is close: it is an infinite loop space after group completion. The category of summable functors from a set with $n$-elements to $C$ is a model for the $n$-fold power $C^n$, that is suitable for constructing a delooping (of the group completion) of $BC$.

One way to get a symmetric monoidal category is to start with a category with coproducts and then take its full subcategory of isomorphisms. For some reason it turns out that some of the most interesting infinite loop spaces are obtained in this way: the first space of the sphere spectrum colim$_n Omega^nS^n$ is obtained from the category of finite sets and bijections. The $K$-theory space $mathbb Z times BO$ is obtained from the category of vector spaces and isomorphisms, etc. However, there also are interesting examples of infinite loop spaces that are obtained as the group completion of a symmetric monoidal category where not all morphisms are isomorphisms. Various categories of cobordisms come to mind.

reference request – Book recommendation: The Gamma function

I don’t know if Math Stackexchange is for such questions (probably, I can ask such questions since there is a tag on it) but I want a book on the Gamma function, that is similar to, say, Titchmarsh’s Theory of the Riemann zeta function. I want a book that studies the gamma function in depth. So please recommend a book on the Gamma function that not just lists some properties of the gamma function. I can search Google for it, but I am not sure.

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.