Tagged Questions

102 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
37 views

Series expansion of incomplete gamma function ratio

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 ...
127 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
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100 views

Behavior of $\Gamma(z)$ as $\text{Im} (z) \to \pm \infty$

In a paper I'm reading it states that $\displaystyle |\Gamma(z)| = |\Gamma(a+ib)| \sim \sqrt{2 \pi} |b|^{a-\frac{1}{2}} e^{-|b|\frac{\pi}{2}}$ as $\displaystyle|b| \to \infty$. How is that derived ...
140 views

Using the gamma function as an upper and lower bound to the logarithm of a factorial function.

I am trying to find an upper and lower bound for the following function: $$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$ where ...
146 views

Upper bound of function including Pochhammer symbol

How can I find the upper bound of $$\left\vert\frac{(c+1/2+\lambda)_{n}}{\lambda^{n}}\right\vert,\quad\text{where}\quad(c+1/2+\lambda)_{n}=\frac{\Gamma(c+1/2+\lambda+n)}{\Gamma(c+1/2+\lambda)}$$ and ...