Linked Questions

0
votes
0answers
28 views

What is the asymptotic “inverse” of the factorial? [duplicate]

Take the following algorithm ...
0
votes
0answers
27 views

How to solve equations with Stirling Approximation? [duplicate]

As we all know Stirling Approximation is giving us an approximate value of factorial, aka $\Gamma(x + 1)$. $\sqrt{2\pi n}(\frac{n}{e}) \approx n!$ But what if we have equations with factorials. In ...
15
votes
1answer
2k views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation $$n!...
4
votes
2answers
918 views

What approximations for the Gamma function's inverse appear to work 'best'?

So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods: Lagrange inversion theorem: $$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to a}\frac{(z-\Gamma(a)...
9
votes
2answers
134 views

On Ramanujan's approximation, $n!\sim \sqrt{\pi}\big(\frac ne\big)^n\sqrt [6]{(2n)^3+(2n)^2+n+\frac 1{30}}$

Over here I discovered that Ramanujan gave the following factorial approximation, better than Stirling's formula: $$n!\sim \sqrt{\pi}\left(\frac ne\right)^n\sqrt [6]{(2n)^3+(2n)^2+n+\frac 1{30}}$$ ...
3
votes
3answers
67 views

Solving for an unknown $c$ in relation to a ratio of gamma functions

I have been working with ratios of gamma functions and I am surprised how difficult it is to make even elementary conclusions. I am hoping it is just the learning curve. Consider the following ...
0
votes
2answers
77 views

Solving $n! = 10^6$ for $n$

I am trying to solve $n! = 10^6$ for $n$. I thought to do this using the gamma function: $$(n - 1)! = \Gamma(n) = \int_0^\infty x^{n - 1}e^{-x} \ dx$$ So I have that $$\Gamma(n + 1) = \int_0^\...
1
vote
1answer
62 views

Looking for the inverse of the hyperfactorial function

As a small part in a statistical thermodynamics project, I need to compute the inverse of the hyperfactorial function. So,as I wrote it, I need to find the zero of function $$f(x)=\log (H(x))-k$$ for ...
-1
votes
1answer
46 views

A way to solve equations of the form $(\frac{n}{a})! = b$?

I found myself today needing the natural solutions for $n$ satisfying this equation: $(\frac{n}{5})! = 2$ In this case I "got lucky" and was able to guess that 10 is a suitable solution. ...
0
votes
3answers
50 views

Integral of sinc function over bounded interval

I am trying to determine whether the integral $$\int_0^1 \frac{\sin x}{x} dx$$ can be calculated analytically. I am aware of the definition of the sine integral function $\text{Si}(x)$, but I haven't ...