# Tagged Questions

89 views

132 views

### How to relate this integral for the $\Gamma$ function to the defining integral of the $\Gamma$

In another question of mine at mse I had the detail of the two assumed identities $$C_{b,p} = \int_0^\infty b^{x^p} dx$$ and $$C_{b,p}={n! \over (- \beta)^n}$$ for some $\small n \in \mathbb N$ ...
550 views

### How to calculate these gamma functions?

Equation : $$\int _{0}^{\infty }x^{n}e^{-x}dx=n!=\Gamma(n+1)$$ 1) $$\int _{0}^{1}x^{2}\left( \ln \dfrac {1} {x}\right) ^{3}dx$$ 2) $$\int _{0}^{1}\sqrt[3] {\ln x}dx$$ Hint : $$x=e^{u}$$ ...
231 views

### Solve in terms of the Gamma function

Show: \begin{align*} \int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\,\mathrm d x &=\frac{\sqrt \pi}{4}\left(\frac{\Gamma ...
802 views

### Calculate integrals involving gamma function

What are the usual ways to follow in order to solve the integrals given below? \begin{align*} I&=\int_0^1 \ln\Gamma(x)\,dx\\ J&=\int_0^1 x\ln\Gamma(x)\,dx \end{align*}
213 views

### Computing $\int_0^{\infty} \frac{ \sqrt [3] {x+1}-\sqrt [3] {x}}{\sqrt{x}} \mathrm dx$

I would like to show that $$\int_0^{\infty} \frac{ \sqrt [3] {x+1}-\sqrt [3] {x}}{\sqrt{x}} \mathrm dx = \frac{2\sqrt{\pi} \Gamma(\frac{1}{6})}{5 \Gamma(\frac{2}{3})}$$ thanks to the beta function ...
226 views

### a gamma integral

any idea how to do this integral ? $$\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt$$ $j$ is a positive integer. $k$ is a constant - not necessarily ...
I know that the Gamma function with argument $(-\frac{1}{ 2})$ -- in other words $\Gamma(-\frac{1}{2})$ is equal to $-2\pi^{1/2}$. However, the definition of $\Gamma(k)=\int_0^\infty t^{k-1}e^{-t}dt$ ...
I have the following integral $$\int_0^{+\infty} t^{z-1} e^{-t} \frac1{(kt + 1)^s}\mathrm dt$$ where $k>0, s > 0$. How would you suggest to solve it? Without $\frac1{(kt + 1)^s}$ it would be ...