2
votes
1answer
59 views

Integral involving Gamma Function

I am solving the following integral: $$ \int_{-1}^{K} u^B e^{-u} du $$ The solution of the integral is a lower incomplete Gamma Function if -1 is replaced with 0. Can anybody help me in solving the ...
0
votes
1answer
49 views

Integrate the integral. Where $({x_1},{y_1}),({x_2},{y_2})\in[0,a] \times[0,b],x_1<x_2,y_1<y_2$

What is the integration of the below integral? $\|N(u)(x_1,y_1)-N(u)(x_2,y_2)\|\le\|\mu(x_1,y_1)-\mu(x_2,y_2)\|+L_1\|u(x_1,y_1)-u(x_2,y_2)\|+\|\dfrac{L_2}{\Gamma (r_1)\Gamma ...
1
vote
1answer
54 views

Evaluating $\dfrac{1}{\Gamma (r)}\int_{0}^{x}(x-t)^{\alpha -1}t^{\lambda}dt$ [closed]

How can I evaluate the following integral $$\frac1{\Gamma(r)}\int_0^x(x-t)^{\alpha-1}t^\lambda\ dt$$
1
vote
2answers
74 views

Compute $\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$ as a function of $\Gamma(n)$

Is it possible to compute this integral $$\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$$ as a function of complete gamma $\Gamma(n)$. If possible, I'm looking for a closed form solution. Thanks!
0
votes
1answer
24 views

Two definitions of the Incomplete Gamma Function - are they equivalent?

From Loss Models, 4th ed., by Klugman et al.: Definition 5.5 The incomplete Gamma function with parameter $\alpha > 0$ is denoted and defined by $$\Gamma\left(\alpha ; x\right) = ...
0
votes
0answers
46 views

Solving integral that contain upper incomplete gamma function, exponential, and powers

I have this integration formula; $ f=\int\limits_{0}^{\infty}\frac{e^{-b~z}}{\sqrt{z}} \Big(\frac{\beta}{\beta+z}\Big)\Big(\frac{\beta+z}{z}\Big)^L ...
0
votes
1answer
26 views

Integral / Gamma Expectation

I would like to solve the following integral, $\int_{0}^{\infty}\frac{\phi}{a+b\phi} \phi^{c-1}e^{-d\phi}d\phi$. Note $\phi \sim Ga(c,d)$ is a gamma distributed random variable and the integral can ...
0
votes
1answer
45 views

Definite integral

So I was playing around with Euler's Reflection Formula ($\Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin(\pi s)}$), trying to prove it with calculus, and was able to reduce $$ ...
0
votes
1answer
130 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
1
vote
4answers
111 views

Some gamma function questions…

I have shown that $\Gamma(a+1)=a\Gamma(a)$ for all $a>0$. But I'd also like to show the following 2 things: 1) Using the previous fact, I'd like to show that $\lim_{a \to 0^{+}}a\Gamma(a) = ...
7
votes
4answers
161 views

Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$ I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0. $$ I am ...
4
votes
2answers
157 views

Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$

$$ I:=\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}. $$ Thank you. The Gamma function is given by $\Gamma(n)=(n-1)!$ and its integral representation is $$ \Gamma(x)=\int_0^\infty t^{x-1} ...
1
vote
2answers
68 views

Evaluating $\frac{d}{dx}\int_3^{x^2}e^{t^3}dt$

$\frac{d}{dx}\int_3^{x^2}e^{t^3}dt$ I suppose I don't fully understand the notation used within this problem. Using the second fundamental theorem of calculus: $\int_a^b f(x)dx = F(x)\bigr|_a^b = ...
2
votes
1answer
32 views

Getting from a product of gamma functions to a fraction answer

I am working on an assignment question for my Advanced Calculus course and am having great difficulty working it out. In order to try and understand this type of question/working, I have found a ...
0
votes
0answers
33 views

Question concerning the gamma function in relation to other holomorphic functions when $Re(\xi) > 0$

Let $f$ be indefinitely differentiable on $\mathbb R$ that has compact support. $\implies f$ belongs to the Schwartz space. Consider: $$I(\xi) = \frac1{\Gamma(\xi)} \int_0^\infty f(x)x^{-1+\xi}dx$$ ...
0
votes
1answer
45 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct a probabilistic model, I have tried different methods of approach, by parts, substitution, but to no avail. Any help with this would be greatly appreciated!
1
vote
1answer
42 views

Proving that $\Gamma (x) = \int_{0}^{1} \left( \ln \left(\frac{1}{u} \right) \right)^{x-1} du$

I want to prove that $$ \Gamma (x) = \int_{0}^{1} \left( \ln \left(\frac{1}{u} \right) \right)^{x-1} du $$ I start with $$ \int_{0}^{1} \left( \ln \left(\frac{1}{u} \right) \right)^{x-1} du = $$ ...
1
vote
1answer
62 views

Gamma like integral with power in exponent

I came across an integral in my stat mech book of the form $$\int_0^\infty x^{d-1}e^{x^s}dx$$ The book claims without proof that this is $$\frac 1s\Gamma(d/s)$$ I tried doing a change of variables ...
2
votes
2answers
54 views

Integration by using special functions

$$\int ^{\pi }_{0}\dfrac {dt}{\sqrt {3-\cos t}}$$ How can you solve the following equation by using alpha/gamma functions and putting $$\cos t=1-2\sqrt {u}$$
1
vote
1answer
31 views

Upper incomplete gamma integral

I would like to know whether the following relation is correct or not? $\frac{d}{dz}\Gamma(w,\mu z)= -\mu^wz^{w-1}e^{-\mu z}$, where $\Gamma(w,\mu z)$ is the upper incomplete gamma integral. Can ...
8
votes
2answers
178 views

Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.

$$ \int_{-\pi/2}^{\pi/2}\cos^m\left(x\right){\rm e}^{{\rm i}n x}\,{\rm d}x ={\pi \over 2^{m}}\, {\Gamma\left(1 + m\right) \over \Gamma\left( 1 + \left[m + n\right]/2\right)\ \Gamma\left( 1 + ...
2
votes
2answers
140 views

Integral calculus question relating to particle motion

"A particle of mass m is attracted toward a fixed point 0 with a force inversely proportional to its instantaneous distance from 0. If the particle is released from rest, at distance L, from 0, find ...
0
votes
1answer
166 views

Integral of incomplete gamma function

I am trying to integrate this: $$\int_0^\infty z^{-|M|-1}\,\Gamma(A,z)\;dz$$ where $A$ is a real positive, and note that the power of $z$ is $-|M|-1$, i.e., is forced to be negative real.
1
vote
0answers
46 views

Prove an equation about fractional integral

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Can anyone help me solve this? I really appreciate. If $f$ is continuous on $[0, \infty)$, for $\alpha \gt ...
4
votes
1answer
67 views

Why $dt/t$ in Mellin transform

I've noticed that often when people write the Gamma function $\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,dt$, that they write it like $$ \Gamma(s) = \int_0^\infty t^s e^{-t}\,\frac{dt}{t} , $$ where ...
5
votes
3answers
205 views

Proving $\int\limits_{0}^{\infty}\frac{1-\text{e}^{-x}\cos(ax)}{x^{r+1}}\operatorname d\!x = \frac{\Gamma(1-r)}{r}(1-a^2)^{r/2} \cos(r \arctan(a))$

does anyone have an idea or a guess how to prove the following equation: $$\int\limits_{0}^{\infty}\frac{1-\text{e}^{-x}\cos(ax)}{x^{r+1}}\operatorname d\!x = ...
2
votes
1answer
68 views

How do I integrate $x^{\frac{3}{2}}e^{-x}$ from 0 to inf?

I have to evaluate the following expression : $$\int^{\infty}_{0} x^{\frac{3}{2}}e^{-x}$$ Wolfram|Alpha evaluates to $\frac{3\sqrt{\pi}}{4}$. I don't see how we got there. A hint would be helpful. ...
0
votes
2answers
67 views

resolving integral using gamma function

My book solves this integral \begin{equation} \int_0^{\infty} y^3e^{-ay} dy \end{equation} using gamma function as \begin{equation} \frac{1}{a^4}\Gamma(4) \end{equation} why is this true?
4
votes
1answer
184 views

An integral representation of the gamma function

An integral representation of the gamma function that would appear to be valid for all complex values of $z$ excluding the integers is $$ \Gamma(z) = \frac{2e}{\sin \pi z}\int_{0}^{\infty} ...
4
votes
2answers
258 views

How to solve this integral: $\int_{-1}^{1} x^k (1-x^2)^{(n/2)-2} \, dx$

How to solve this integral step by step: $$\int_{-1}^{1} (x^k) (1-x^2)^{(n/2)-2}dx=??? $$ In my text book, it shows the result like below: $$\int_{-1}^{1} (x^k) (1-x^2)^{(n/2)-2}dx= ...
0
votes
1answer
63 views

How to solve this gamma integral

Let we have the (p.d.f) of x which is: $$f(x)=\frac {\Gamma{(n-1)/2)}}{\Gamma{(1/2)} \Gamma{(n-2)/2)}}x(1-x^2)^{(n/2)-2}$$ then to find the $E(x) = $$\int_{-\infty}^{\infty} x *(f(x) dx$ ; ...
7
votes
1answer
200 views

Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$

As it says in the title, I'd like to know how to solve the definite integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$. Mathematica gives the answer $\frac{1}{2}\log (2\pi)-1$ but I have no idea how one ...
-1
votes
1answer
145 views

How to solve gamma function integral for 4!

I have been trying to test my knowledge of the gamma function by calculating 1! and 4!. I got the right result for 1! but I cannot get 4! analytically. To be more specific, I think I am not ...
2
votes
1answer
71 views

Some more parameter integrals

It seems that the following formulas hold : $$\int_{0}^{\infty} \frac{1}{\sqrt{x^{2n}+1}} dx = \frac{\Gamma(\frac{n-1}{2n})\Gamma(\frac{2n+1}{2n})}{\sqrt\pi}$$ for any integer $n > 1$ and ...
1
vote
1answer
72 views

Let $a_n = \int\limits_{0}^{n} e^{-x^4} dx$. Does $\{ a_n \}_{n \rightarrow \infty}$ converge?

Let $a_n = \int\limits_{0}^{n} e^{-x^4} dx$. Does $\{ a_n \}_{n \rightarrow \infty}$ converge? $\{ a_n \} =\{ \int\limits_{0}^{1} e^{-x^4} dx, \int\limits_{0}^{2} e^{-x^4} dx, ..., ...
4
votes
6answers
204 views

How to integrate $\int_{0}^{a}x^{n-1}e^{-x}dx$

We know that $$\int_{0}^{\infty }x^{n-1}e^{-x}dx = \Gamma (n)$$ But how do we integrate this? $$\int_{0}^{a}x^{n-1}e^{-x}dx$$
3
votes
1answer
101 views

Interesting integral involving $\Gamma (z)$.

Find the value of $$\int_0^\infty t^{x-1}e^{-\lambda t \cos(\theta)} \cos(\lambda t \sin (\theta)) dt$$ where $\lambda >0$, $x>0$, and ${-1\over 2}\pi < \theta < {1\over 2}\pi$ in terms of ...
3
votes
1answer
85 views

Expressing integral as gamma function

I was trying to compute the expected value $E[X^k]$ for a Weibull distribution, and I encountered this integral $$\int_{-\infty}^\infty t^{b+k-1}e^{-ct^b}dt$$ where $k>0$ is an integer and ...
3
votes
1answer
115 views

Gamma function in the sight of Lebesgue and Riemann integration.

I am taking a somewhat hard measure theory course and I was asked to prove this: a) Let $\alpha > 0$ be a real number. Prove that $$\Gamma(\alpha):=\int_0^\infty e^{-x}x^{\alpha-1}dx$$ exists. ...
12
votes
0answers
495 views

The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \ \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \ \mathrm dx = \frac{z}{2} \log (2 \pi) + ...
6
votes
2answers
138 views

Gamma Type Integral

I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ...
1
vote
0answers
108 views

Limiting behavior of an integral involving incomplete Gamma function

I am wondering about the limiting behavior as $k\rightarrow\infty$ of the following integral: $$I(k)=\frac{2^{-k/2}}{\Gamma(k/2)}\int_{f(k)}^\infty ...
1
vote
1answer
31 views

Relation between two gamma functions

Does anyone know the relation between these two gamma functions? 1st) Gamma[1 + c, a (1 + b)] 2nd) Gamma[c, a (1 + b)] The question is: may I write the 1st like the 2nd times something? thank you ...
1
vote
0answers
33 views

Show that $\sum\limits_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$. [duplicate]

Prove that for $x,y$ positive integers, $$\sum_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$$ One way is to use the beta-gamma functions relation: ...
1
vote
3answers
165 views

I'm looking for several ways to prove that $\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$

I'm looking for several ways to prove that $$\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$$ for $-2< Re(m)< 0$
5
votes
2answers
87 views

Using $\Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! n^z}{z(z+1)(z+2)\ldots(z+n)} \right]$ to prove the Weiestrass product.

I was searching the web quite thoroughly in the last two days. I was in paralytically looking for a rigorous proof using $$ \Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! ...
16
votes
6answers
547 views

Show that $\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =-\frac{\pi^2 \sqrt{2}}{16}$

I could prove it using the residues but I'm interested to have it in a different way (for example using Gamma/Beta or any other functions) to show that $$ ...
4
votes
1answer
134 views

Integrate the beta function $B(x,y)$ over the unit square.

I was looking a bit around on the site and disovered (to my surprise) that $$ \int_0^1 \log \Gamma(x+t)\,\mathrm{d}x = \log \Bigl(\sqrt{2\pi}\Bigr) - t + t \log t $$ and using this and the fact ...
13
votes
7answers
639 views

Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. [duplicate]

Can anyone give me a hint to evaluate this integral? $$\int_0^\infty \frac{dx}{1+x^4}$$ I know it will involve the gamma function, but how?
5
votes
1answer
147 views

Integration gamma and beta: $\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$

How can we evaluate the following integral? $$\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$$ I can't find anything to substitute because all of the trigonometric identities are in square form...