Gamma
n!¶
\[ n! = ? \]
\[ n! = n(n - 1)(n - 2)... 1 \]
\[ \frac{d}{dx} x^n = n x^{n - 1} \]
\[ \frac{d^n}{dx^n} f(x) = : \frac{d}{dx} ( \frac{d^{n - 1}}{dx^{n - 1}} f(x) ) \]
\[ \frac{d^1}{dx^1} f(x) = : \frac{d}{dx} f(x) \]
\[ \frac{d^2}{dx^2} x^n = n(n - 1) x^{n - 2} \]
\[ \frac{d^a}{dx^a} x^n = \frac{n! x^{n - a}}{(n - a)!} \]
\[ \frac{d^n}{dx^n} x^n = \frac{n! x^{n - n}}{(n - n)!} = n! \]
\[ \frac{d^n}{dx^n} \frac{1}{x} = \frac{(-1)^n n!}{x^{n - 1}} \]
\[ \frac{d^n}{dx^n} (-\frac{1}{x}) = \frac{n!}{(-x)^{n + 1}} \]
\[ \text{if} \frac{n!}{(-x)^{n + 1}} \text{is a function of x and x can be a constant, then} \frac{d^n}{dx^n} (-\frac{1}{x}) \text{is not. so} \frac{n!}{(-x)^{n + 1}} \text{at} x = -1$ \text{is} n! \]
\[ \text{problem solved!... but that is a bit too many derivitaves...} \]
\[ \text{so an example of something easy to differentiate is } e^x \]
\[ \text{this line of text is in memorial of spamming quad quad quad quad quad quad quad quad insead of using the /text feature} \]
\[ \frac{d}{dx} e^x = e^x \]
\[ \frac{\partial}{\partial t} e^{xt} = x e^{xt} \]
\[ \frac{d}{dx} (\int f(x) dx) = : f(x) \]
\[ \int_{a}^{b} f(x) dx = : \int f(b) - \int f(a) \]
\[ \int f(x) = \int_{a}^{x} f(t) dt \]
\[ \int_{0}^{\infty} f(x) dx = : (\lim_{x \to \infty} f(x)) - f(0) \]
\[ \int e^{xt} dt = \frac{e^{xt}}{x} \]
\[ \int_{0}^{\infty} e^{xt} dt = \frac{e^{\infty x}}{x} - e^{0x} = -\frac{1}{x}, x<0 \]
\[ \frac{d^n}{dx^n} \int_{0}^{\infty} e^{xt} dt = \frac{d^n}{dx^n} (-\frac{1}{x}), x<0 \]
\[ \int_{0}^{\infty} \frac{\partial^n}{\partial x^n} e^{xt} dt = \frac{n!}{(-x)^{n + 1}}, x<0 \]
\[ \int_{0}^{\infty} t^n e^{xt} dt = \frac{n!}{(-x)^{n + 1}}, x<0 \]
\[ \int_{0}^{\infty} t^n e^{-t} dt = n! \]
\[ \Gamma (x) = : \int_{0}^{\infty} t^{x - 1} e^{-t} dt \]
problem solved! but this has no constraint that \(n\) is an integer, so I will use \(x\) now that it is freed up from the formula, and the factorial has its own definition so the \(4 \mu l(a)\) la has been demoted to \(\Gamma (x + 1)\)
(four-mu-la)
.¶
.
.
.
.
.
.
.
.
.
.
.
.
.
. you found it!