How to show $$\frac{\Gamma(k+b)}{\Gamma(k)\Gamma(b+1)} = \frac{k^b}{\Gamma(b+1)}\left(1+O\!\left(\frac1k\right)\right),$$ where $b \in [0,1]$ and $k \in \mathbb{R}$?

For positive integer $n$, an integral of the form $\int_0^\infty x^n e^{-ax}\,\mathrm{d}x$ will reduce to $\frac{n!}{a^{n+1}}$. This can be shown by successively applying integration by parts and observing that the $\left.f(x)g(x)\right|_0^\infty$ terms vanish. The first such term $f(x)g(x)$ is $\left.-\frac{1}{a}e^{-ax}x^n\right|_0^\infty$. It is clear that an exponential will “beat” any power function as $x$ tends to […]

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. My attempts were to use the “By Parts” rule, when I realized that this is the famous Gamma function. There are several sources on internet which give a way […]

Following @mercio’s comments, I’ve rewritten my question in terms of zeros instead of saddles. Also, after more careful consideration, I’ve decided that perhaps the path I seek might not depend on the presence of poles or choice of end points, but I’m not sure. In any case, please help me prove or disprove the following […]

I am interested in the series expansion of: $$S(k)=\frac{\Gamma(k+1,a)}{k!},$$ around $k=\infty$ where $\Gamma(x,z)$ is the incomplete gamma function and $a$ is some positive constant. In particular, I would like to know how quickly this ratio converges to 1. How to prove asymptotic limit of an incomplete Gamma function shows that: $$1-S(k)\leq \left(\frac{e^a-1}{e^a}\right)^{k+1},$$ but is it […]

Can we show that the following function is integrable \begin{align} f(t) =\frac{2^{\frac{it+1}{1.5}}}{2^{\frac{it+1}{2}}} \frac{\Gamma \left( \frac{it+1}{1.5} \right) }{\Gamma \left(\frac{ it+1}{2} \right)}, \end{align} where $t \in \mathbb{R}$ and $i=\sqrt{-1}$. That is can we show that \begin{align} \int_{-\infty}^{\infty} |f(t)| dt<\infty. \end{align} I was wondering if Stirling’s approximation can be used, since this is a complex case? Note if […]

I have reduced this problem (thanks @Mhenni) to the following (which needs to be proved): $$\prod_{k=1}^n\frac{\Gamma(3k)\Gamma\left(\frac{k}{2}\right)}{2^k\Gamma\left(\frac{3k}{2}\right)\Gamma(2k)}=\prod_{k=1}^n\frac{2^k(1+k)\Gamma(k)\Gamma\left(\frac{3(1+k)}{2}\right)}{(1+3k)\Gamma(2k)\Gamma\left(\frac{3+k}{2}\right)}.$$ As you see it’s quite a mess. Hopefully one can apply some gamma-identities and cancel some stuff out. I have evaluated both products for large numbers and I know that the identity is true, I just need to learn […]

Is there any relation between the limiting behaviour of $\Gamma({\epsilon})$ and $\Gamma(-1+{\epsilon})$? I have seen the relation such as $\Gamma(-1+{\epsilon})$ $=$ $\Gamma({\epsilon})/(-1+{\epsilon})$. I think it is basically wrong? But does there exist such a similar relation?

I know there are several approximations of the Gamma function that provide decent approximations of this function. I was wondering, how can I efficiently estimate specific values of the Gamma function, like $\Gamma (\frac{1}{3})$ or $\Gamma (\frac{1}{4})$, to a high degree of accuracy (unlike Stirling’s approximation and other low-accuracy methods)?

I’ve already proved that $$\Gamma (n)= (n-1)!$$ but I don´t really know what else to do to verify that $\Gamma$ is an extension of the factorial function for real numbers (positive) Thank you! And I´m sorry for my language, I am Spanish, so thank you again for trying to understand me.

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