0
votes
1answer
130 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
1
vote
2answers
68 views

Evaluating $\frac{d}{dx}\int_3^{x^2}e^{t^3}dt$

$\frac{d}{dx}\int_3^{x^2}e^{t^3}dt$ I suppose I don't fully understand the notation used within this problem. Using the second fundamental theorem of calculus: $\int_a^b f(x)dx = F(x)\bigr|_a^b = ...
2
votes
1answer
29 views

Calculate the derivative of $Γ(z,v)$ with respect to $z$

Let $Γ(z,v)=∫_{v}^{+∞}t^{z-1}e^{-t}dt$ be the incomplete $Γ$-function. My question is: Calculate the derivative of $Γ(z,v)$ with respect to $z$.
1
vote
1answer
71 views

How to derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n+ 1)$?

How can you derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n + 1)$? I have tried checking Wolfram Alpha for a step-by-step solution, but none is given. Moreover, of what is the second function, ...
2
votes
1answer
161 views

differentiate log Gamma function

I am working with the log of negative binomial distribution $NegBin(r,p)$. I need to differentiate the following with respect to $r$ such that at the end, I am NOT left with $r$ in factorial form. ...
1
vote
2answers
82 views

Why do we have $\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt$?

Why is the following representation true? $$\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt,$$ where $\psi^{(m)} (z)$ denotes the Polygamma function.
1
vote
1answer
49 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
116 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
87 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
91 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
335 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 ...