Gamma function Definition and 125 Threads

I was searching for derivative of factorial. Many say that gamma function is the derivative of the factorial. Is that true because I searched about gamma function and it doesn't say anything like that. Thanks a lot

Homework Statement The absolute value of the gamma function \Gamma (x) that is defined on the negative real axis tends to zero as x \to - \infty . Right? But how do I prove it? Homework Equations The Attempt at a Solution I've tried to use Gauss's Formula...

Now, I already know the complement formula and I also know a proof that can be presented using the Euler product forms of the Gamma function. What I am curious about is, can one obtain the formula using the integral forms of the Gamma and Beta functions? I will show my work so far. I started...

Can someone help me to expand the following gamma functions around the pole ε, at fisrt order in ε \Gamma[(1/2) \pm (ε/2)] where ε= d-4

Homework Statement Evaluate the intergrals: a) integral of 3^(-4*z^2) dz from 0 to infinity b) integral of dx/(sqrt(-ln(x))) from 0 to 1 c) integral of x^m * e^(-a*x^n) dx from 0 to infinity Homework Equations gamma(n) = integral of e^(-w) * w^(n-1) dw from 0 to infinity The...

Homework Statement The gamma function, which plays an important role in advanced applications, is defined for n\geq1 by \Gamma(n)=\int_0^{\infty} t^{n-1}e^{-t}dt (a) Show that the integral converges on n\geq1 (b) Show that \Gamma(n+1)=n\Gamma(n) (c) Show that \Gamma(n+1)=n! if n\geq1 is an...

I have a gamma function in the form of Gamma(s-1) where s can take only positive values. How can I plot this function for different value of s using Mathematica ?

Homework Statement \int_0^∞ x^2exp(-x/2) dx Homework Equations The Attempt at a Solution Using u substitution: u = x/2 du = 1/2 dx \int_0^∞ 4u^2exp(-u) du*2 = 8 \Gamma(3) = 8*3! = 48 But the correct answer is 16 when I plug it in Wolfram's definite integral...

Homework Statement Prove by induction that gamma(v+1)(v+1)(v+2)...(v+k)=gamma(v+k+1) for k=1,2,3... Homework Equations Really just using the relation x*gamma(x)=gamma(x+1) The Attempt at a Solution for a basis gamma(v+1)(v+1)=gamma(v+1+1) so holds for k = 1 inductive...

\Gamma(x)=\int^{\infty}_0t^{x-1}e^{-t}dt \Gamma(\frac{1}{2})=\int^{\infty}_0\frac{e^{-t}}{\sqrt{t}}dt= take t=x^2 dt=2xdx x=\sqrt{t} =\int^{\infty}_0\frac{e^{-x^2}}{x}2xdx Why here we can here reducing integrand by x?

If the Gamma function \Gamma (z) = \int_0^{\infty} t^{z-1} e^{-t}\;dt only converges for \text{Re}(z)>0 then why is, for example, \Gamma (-1+i) defined when clearly \text{Re} (-1+i)<0 ?

How do I calculate the integral \int_{ix}^{i\infty} e^{-t} t^{-s-1}dt, where x>0, s>0? Mathematica gives \Gamma(-s,ix), where \Gamma(\cdot,\cdot) is the incomplete gamma function, but I am not sure how to justify this formally.

is there a way to explicitly express the chi-squared inverse function? when programming it, I have had to resort to a guessing system where I find a chi value that is too low and too high, and evaluate the chi-squared CDF to reset the high and low points iteratively until it is within a...

In some exercises I've stumbled upon a function which is denoted \gamma_{m}(n) with m,n natural. I've no idea what is the definition of the function and could not infer from the exercises. Searching google yielded nothing, as it kept suggesting me the OTHER Gamma function. Can anyone here help...

Hello, In our course of complex analysis we proved that the gamma function, \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm d t for 0 < Re(z), is analytical. We did this by defining f_{\epsilon,R}(z) = \int_\epsilon^R t^{z-1} e^{-t} \mathrm d t about which we can prove that it is analytical...

Hi all, I'm looking at the http://en.wikipedia.org/wiki/Gamma_function#General" for the gamma function, and it lists equations for the gamma function of half-integer orders (i.e. gamma(0.5+n) and gamma(0.5-n)). But, it doesn't list a reference as to where this equation comes from. Does...

Gamma Function on negative Fractions! If we take a look at the Gamma Function and evaluate the integral by parts then we will get infinity in the first step of Integration by Parts eg: Integral e^-1*x^-5/3 Limits being 0 to Infinity as usual! If we try to integrate this we will get...

Homework Statement In attachment The Attempt at a Solution I can't see the link in the steps, a little help pls.

Homework Statement \int xe^{-x^{3}}dx from 0 to infinity Homework Equations \Gamma = \int x^{p-1}e^{-x} from 0 to infinity The Attempt at a Solution my problem is I'm not sure what i am supposed to do with the exponent of 3 on the e, because it seems to affect the answer...

***Please skip ahead to post # 6 where I have better formulated my question. Thank you! *** I have an equation AY'' + B\eta^2Y'=0 \qquad(1) where A and B are known constants and Y is a function of η. By using the substitution X = Y' I have reduced the problem to a first order ODE of...

Hello, I need help proving this: http://mathworld.wolfram.com/images/equations/GammaFunction/Inline177.gif = http://mathworld.wolfram.com/images/equations/GammaFunction/Inline179.gif

I was just curious as to how I can show the following properties of the Gamma Function, they came up in some lecture notes but were just stated? Notation: G(z) = Gamma function 2^(z) = 2 to the power of z I = Integral from 0 to infinity (1) G(z)*G(1-z) =...

Hi, I am interested in performing the following integration: \int _{-\infty }^{\infty }\Gamma\left[k,\frac{x+u}{v}\right]e^{-\frac{(x-m)^2}{2\sigma ^2}}dx . I would appreciate anyone's help. I have been trying to do it in Mathematica but it runs out of time returning the same integral...

Homework Statement

I know that the gamma function (from 0 to infinity): \int e-t tx-1 dt = \Gamma(x) and that the relation exists... \int e-ut tx-1 dt = 1/ux \Gamma(x) Now for the lower bound incomplete gamma function... I see that from http://people.math.sfu.ca/~cbm/aands/page_260.htm (equation...

Homework Statement Why is the equality below true? \Gamma(n) = (n-1)! Where \Gamma(n) = \int^{\infty}_{0} x^{n-1} e^{-x}dxHomework Equations The Attempt at a Solution I've read the article on wikipedia but I cannot understand it. Is there any special properties in calculus that I must know...

What is incomplete gamma function? How are they evaluated? thks for any ans.

I am using a particular form of the incomplete gamma function (which I have never seen before) in my probability course. It is denoted: F(x;\alpha) = \int_0^x \frac{y^{\alpha - 1}e^{-y}}{\Gamma(\alpha)}\,dy\qquad(1)Question 1 Why the bounds in terms of 'x' ? I am just a little confused by...

What are practical applications of the gamma function?

I want to verify the procedure of finding .\Gamma(n+p+1). with p =-ve. This is usually found in Bessel's equation. It is well talked about if p=+ve. But books I have don't even talking about in general how to find the series representation when p=-ve. I worked this out and I want to verify with...

Homework Statement This is actually part of a probability problem I'm thinking about. I'm trying to find the nth moment of a certain random variable in terms of the gamma function, which is basically equivalent to solving the following integral or expressing it in terms of the gamma function...

I have come across this expression in some notes \Gamma (z) = \frac{1}{z} \prod \frac{(1+ \frac{1}{n})^{z}}{1+ \frac{z}{n}} Do you think it's accurate? I have some doubts because I have looked for it on wokipedia, and I couldn't find it.

Homework Statement I (2n,m) = Integral cos^(2n)O sin^(m)O cosO dO limits are 0 to 2pi and O = theta 0.4/3 = 0.1333 show that I 2n,m = 2n/ m+1 (I2n-2, m+2) Homework Equations I really have no idea how to work with this problem. It is under Gamma function of the instructors...

k*Γ((n-1)/2 + 1)=Γ(n/2 + 1) I need to solve for k, and I'm having some difficulty manipulating the gamma function to obtain my desired result. Any properties, hints or help would be greatly appreciated.

Can you show me how to get the series representation of \Gamma(n-3/2+1)? For example \Gamma(n+3/2+1)=\frac{(2n+3)(2n+1)!}{2^{2n+2}.n!}. I cannot figure out how to write a series with: n=0 => \Gamma(0-3/2+1)= -2\sqrt{\pi} n=1 => \Gamma(1-3/2+1)= \sqrt{\pi} n=2 =>...

Homework Statement Define the function: f(x)= The integral from 0 to infinity of t^(x)e^(-t)dt. Find f(3), f(4) and f(5). Notice anything? Homework Equations N/A The Attempt at a Solution I assume that I start by finding the integral of f(x). I used wolfram alpha and found that...

\[Gamma][v_] := 1/Sqrt[1 - (v)^2] plot1 = Plot[\[Gamma][v], {v, -.99, .99}, PlotStyle -> {Thick, RGBColor[0.6, 0, 0]}, PlotRange -> All] \.08 plot2 = Plot[0, {v, -.99, .99}, PlotRange -> All] \.08 Show[plot1, plot2, PlotRange -> All, AxesLabel -> {"\!\(\*FractionBox[\"v\"...

A very vague question: What is the derivative of the gamma function? Here's what I've got, using differentiation under the integral. Can anybody tell me if I'm on the right track? What does my answer mean? \Gamma(z) = \int_0^{\infty} t^{z - 1} \: e^{-t} \; dt The integrand can be...

\Gamma(z)=\int^{\infty}_0x^{z-1}e^{-x}dz z\in\mathhad{C} In which problems in statistical physics we need gamma functions of complex argument? I don't know how to calculate \Gamma(i) for exaple?

Hello all, Is there any way to make the Gamma function \Gamma[z] defined at z=0? Because in my calculations, I have a single case among many other cases where the argument of the gamma function equals zero. Regards

I spend some time studying special functions recently. I found two definitions of gamma function, one in form of integral and the other in form of infinite products, and I cannot prove of their equivalence. I found the definition in infinite product form important in proofing many properties of...

I need to find a way to sum/ a closed form representation for: \sum^{N}_{n=1}\frac{\Gamma(n-s)}{\Gamma(n+s)} 0<Re(s)<1 Thanks for the help in advance.

I need to show whether or not: |\gamma(s,1)| converges for 0<Re(s)<1. Does anyone know of an expansion for this that would prove convergence?

\Gamma(n) = (n-1)! Corrrect?

Homework Statement I read in a paper that: \Gamma\left(c,\,d\frac{x+e}{x-y}\right) = (c-1)!\,exp\left[-d\frac{y+e}{x-y}\right]\,exp[-d]\,\sum_{k=0}^{c-1}\,\sum_{l=0}^k \frac{d^k}{k!}{k\choose l}\left(\frac{y+e}{x-y}\right)^l Homework Equations But the incomplete gamma function...

Could you consider the gamma function to be a closed form representation? If I could express a numerical series in terms of the gamma function, would it be considered a closed form representation?

How to prove that this formula is correct: \lim_{x\to\infty} \frac{\Gamma(x+1)}{\sqrt{2\pi x}\big(\frac{x}{e}\big)^x} = 1 I have seen a proof for this: \lim_{n\to\infty} \frac{n!}{\sqrt{2\pi n}\big(\frac{n}{e}\big)^n} = 1 but it cannot be generalized easily for gamma function. The proof...

\Gamma(z) = \int\limits_0^{\infty} t^{z-1} e^{-t} dt I can see that if \textrm{Re}(z)>0, then the integral converges, and that if \textrm{Re}(z)\leq 0 and \textrm{Im}(z)=0, then it diverges. However, I found the case \textrm{Re}(z)\leq 0 and \textrm{Im}(z)\neq 0 more difficult. t^{z-1} =...

can anybody help me to proove gamma(n+1/2)=(2n)!*sqrt(pi)/((4^n)*n!) thank you

1. Numerically approximate \Gamma(\frac{3}{2}). Is it reasonable to define these as (\frac{1}{2})!? 2. Show in the sense of question 1. that (\frac{1}{2})! = \frac{1}{2}\sqrt{\pi} at least numerically. How am i supposed to attempt this numerically? given that i do not know additional...