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### Definite integral formula from wikipedia

I need to solve a certain definite integral, and several places (for example Wikipedia) I've come across the following formula: ...
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The original function was defined as $f(z)=y=\left( \Gamma \left( k,{\frac {z-b}{\theta}} \right) -\Gamma \left( k,{\frac {z-a}{\theta}} \right) \right) \left( \Gamma \left( k \right) \right) ... 2answers 145 views ### Resummation of the Series I wonder if there is a way to get resummation of this series? By this way , i am trying to get the integral representation of this series, it could be by Gamma function. $$\sum _{k=0}^{\infty} \left ... 3answers 261 views ### Proof that Γ'(1) = -γ? I know that Γ'(1) = -γ, but how does one prove this? Starting from the basics, we have that:$$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$How do we differentiate this? How do we then find that ... 4answers 207 views ### Gamma identity \lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p})dx=\Gamma(p+1) I ran across what appears to be another Gamma identity. Show that$$\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p}) \,\mathrm dx=\Gamma(p+1)=p!$$I tried several different subs and ... 2answers 566 views ### Convergence of \Gamma(p) for 0<p\leq 1 and divergence for p \leq0. Can someone show me a proof or any clear resource about convergence of gamma function for values of p less than zero. If possible I need proofs using integration by parts. My problem evaluating ... 2answers 114 views ### How to integrate this equation all. I am trying integrate this equation(gamma density).$$\int\limits_0^\infty \frac{1}{\sqrt{\left| x-1 \right|}}\exp \left( -\left| 1-x \right| \right) \;dx$$What I have done is split it into 2 ... 1answer 180 views ### Intuitive limit. Integration and \gamma(n,x) It isn't hard to prove that:$$\int_0^x e^{-t} {t^n} dt = n! \cdot e^{-x}\left( e^x-\sum_{k=0}^{n} \frac{x^k}{k!}\right)$$Or put in a different way:$$\int_0^x e^{-t} \frac{t^n}{n!} dt = ... 1answer 199 views ### About integration related to the gamma function I would like to compute the integral $$\int_{0}^{\infty}\frac{1}{\sqrt{2t}}e^{-\frac{1}{2t}}dt$$ which wolfram alpha says that it does not converge. However by letting$x=1/2t$I get ... 2answers 212 views ### Integral form of$\Gamma (x)$While trying to represent the poles and analytic continuation of$\Gamma (x)$, the author uses the following equality: $$\int_{0}^{1}t^{x-1}e^{-t}dt=\sum_{n=0}^{+\infty}\frac{{(-1)}^{n}}{(n+x)n!}.$$ ... 2answers 324 views ### Integration of powers of the$\sin x$I have to evalute $$\int_0^{\frac{\pi}{2}}(\sin x)^z\ dx.$$ I put this integral in Wolfram Alpha, and the result is ... 1answer 138 views ### Coordinate scaling in incomplete gamma function integral I'm faced with the integral $$\mathcal{I} = \int_0^\infty \mathrm d x \; e^{-\beta \, e^x - \mu x} \;,\quad \Re(\beta) > 0 \;.$$ The solution can be looked up. It reads $$\mathcal{I} = \beta^\mu ... 1answer 179 views ### Integral of product of Gamma densities over probability simplex I want to find a simpler form or closed form for the following integral:$$ \int_A \,\prod_{t=1}^T f_{\Gamma(a,\theta_t)}(x_t) \,d\mathbf{x} $$where A is the simplex$$ A = \{\mathbf{x} \in ... 0answers 129 views ### Determining well-definedness for functions How does one determine well-definedness in analytical continuation for$\Gamma(s)\zeta(s)$function? Firstly: $$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$ ... 2answers 198 views ### modified gamma integral 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 ...
The following seems to hold in numerical simulations, is it true? $$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\sqrt{x(1-x)}}=2$$ It's a combination of two known integrals ...