Tagged Questions
1
vote
1answer
30 views
Looking for help understanding the asymptotic expansion of the digamma function
I was recently given an example using this asymptotic expansion of the digamma function where:
$$\frac{d}{dx}(\ln\Gamma(x)) = \psi(x) \sim \ln(x) - \frac{1}{2x} - \frac{1}{12x^2}$$
Here's the ...
4
votes
1answer
111 views
Need help understanding if a function is increasing or decreasing
I am working on understanding the following function:
$$g(x) = \ln\Gamma\left(\frac{x}{4}\right) - \ln\Gamma\left(\frac{x}{5}+\frac{1}{2}\right) - \ln\Gamma\left(\frac{x}{20}+\frac{1}{2}\right) - ...
2
votes
1answer
43 views
The rate of increase of the Gamma Function over real numbers
If
$$ x_1 > x_2 > 0$$
and $$\Delta{x}>0$$
does it follow that:
$$\ln\Gamma(x_1 + \Delta{x}) - \ln\Gamma(x_1) \ge \ln\Gamma(x_2 + \Delta{x}) - \ln\Gamma(x_2)$$
Would it be enough to show ...
1
vote
1answer
58 views
Reasoning about the gamma function using the digamma function
I am working on evaluating the following equation:
$\log\Gamma(\frac{1}{2}x) - \log\Gamma(\frac{1}{3}x)$
If I'm understanding correctly, the above is an increasing function which can be demonstrated ...
5
votes
3answers
265 views
Proof that $Γ'(1) = -γ$?
I know that $Γ'(1) = -γ$, but how does one prove this?
Starting from the basics, we have that:
$$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$
How do we differentiate this? How do we then find that
...
