Gamma
n!¶
Credit: BriTheMathGuy
\[ n! = ? \]
\[ n! = n(n - 1)(n - 2)... 1 \]
\[ \frac{d}{dx} x^n = n x^{n - 1} \]
\[ \frac{d^n}{dx^n} f(x) = : \frac{d}{dx} ( \frac{d^{n - 1}}{dx^{n - 1}} f(x) ) \]
\[ \frac{d^1}{dx^1} f(x) = : \frac{d}{dx} f(x) \]
\[ \frac{d^2}{dx^2} x^n = n(n - 1) x^{n - 2} \]
\[ \frac{d^a}{dx^a} x^n = \frac{n! x^{n - a}}{(n - a)!} \]
\[ \frac{d^n}{dx^n} x^n = \frac{n! x^{n - n}}{(n - n)!} = n! \]
\[ \frac{d^n}{dx^n} \frac{1}{x} = \frac{(-1)^n n!}{x^{n - 1}} \]
\[ \frac{d^n}{dx^n} (-\frac{1}{x}) = \frac{n!}{(-x)^{n + 1}} \]
\[ \text{if} \frac{n!}{(-x)^{n + 1}} \text{is a function of x and x can be a constant, then} \frac{d^n}{dx^n} (-\frac{1}{x}) \text{is not. so} \frac{n!}{(-x)^{n + 1}} \text{at} x = -1$ \text{is} n! \]
\[ \text{problem solved!... but that is a bit too many derivatives...} \]
\[ \text{so an example of something easy to differentiate is } e^x \]
\[ \text{this line of text is in memorial of spamming quad quad quad quad quad quad quad quad instead of using the /text feature} \]
\[ \frac{d}{dx} e^x = e^x \]
\[ \frac{\partial}{\partial t} e^{xt} = x e^{xt} \]
\[ \frac{d}{dx} (\int f(x) dx) = : f(x) \]
\[ \int_{a}^{b} f(x) dx = : \int f(b) - \int f(a) \]
\[ \int f(x) = \int_{a}^{x} f(t) dt \]
\[ \int_{0}^{\infty} f(x) dx = : (\lim_{x \to \infty} f(x)) - f(0) \]
\[ \int e^{xt} dt = \frac{e^{xt}}{x} \]
\[ \int_{0}^{\infty} e^{xt} dt = \frac{e^{\infty x}}{x} - e^{0x} = -\frac{1}{x}, x<0 \]
\[ \frac{d^n}{dx^n} \int_{0}^{\infty} e^{xt} dt = \frac{d^n}{dx^n} (-\frac{1}{x}), x<0 \]
\[ \int_{0}^{\infty} \frac{\partial^n}{\partial x^n} e^{xt} dt = \frac{n!}{(-x)^{n + 1}}, x<0 \]
\[ \int_{0}^{\infty} t^n e^{xt} dt = \frac{n!}{(-x)^{n + 1}}, x<0 \]
\[ \int_{0}^{\infty} t^n e^{-t} dt = n! \]
\[ \Gamma (x) = : \int_{0}^{\infty} t^{x - 1} e^{-t} dt \]
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 \(4 \mu l(a)\) has been demoted to \(\Gamma (x + 1)\)
(four-mu-la)
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. you found it!