Famous mathematician and scientist Carl Gauss developed the Pi function
$$\Pi(x)=\int_{t=0}^\infty t^x e^{-t}\,dt$$
which has the property that
$$\Pi(n)=\prod_{k=1}^{n}\left(k\right)=1\cdot2\cdot3\cdots n=n!$$
for all $n\in\mathbb{N}$. In other words, the Pi function generalizes the factorial operator to the real numbers (and complex numbers also).
Wikipedia says the factorial function grows faster than all exponential functions (and therefore faster than all polynomial functions) but not as fast as hyper-exponential functions, otherwise known as tetration or towering. Well, I would like to know exactly how fast it grows.
What is the rate of growth of the Pi function?
$$\Pi^{\prime}(x)=\frac{d}{dx}\int_{t=0}^\infty t^x e^{-t}\,dt$$
Is the FTC applicable here?
Best Answer
The $\Pi$ function is one of my favourites. Using differentiation under the integral sign,
$$\Pi'(x)=\frac{\text d}{\text dx}\int_0^\infty t^x e^{-t}\,\text dt = \int_0^\infty \frac{\partial }{\partial x}t^x e^{-t}\,\text dt = \int_0^\infty t^{x}e^{-t}\log t\,\text dt \tag{1} = \Pi(x) \Psi (x+1)$$
where $\Psi(x)$ is the digamma function.
Repeated differentiation yields
$$\Pi^{(n)}(x)=\int_0^\infty t^x e^{-t}\log^n t \,\text dt \tag{2}$$
An interesting special case of $(1)$ can be found by letting $x = 0$:
$$\Pi'(0) = \int_0^\infty e^{-t}\log t\, \text dt = -\gamma = -0.577216\dots$$
where $\displaystyle \gamma = \lim_{n \to\infty} 1+\frac{1}{2} + \cdots + \frac{1}{n} - \log n\;\;$ is Euler's Constant.