Mathematical formula

Gaussian integral

∫−∞∞e−x2dx=π \int_{-\infty }^{\infty } e^{-x^2} dx = \sqrt[ ]{ \pi }
∫−∞∞e−ax2dx=πa \int_{-\infty }^{\infty } e^{-ax^2} dx = \sqrt[ ]{ \frac{ \pi }{ a } }
∫−∞∞e−axbdx=1ba−1/bΓ(1b)=a−1/bΓ(1+1b) \int_{-\infty }^{\infty } e^{-ax^b} dx = \frac{1}{b} a^{-1/b} \Gamma \left( \frac{1}{b} \right) = a^{-1/b} \Gamma \left( 1 + \frac{1}{b} \right)

Stirling's approximation

lnn!=nlnn−n+O(ln(n)) \ln n! = n\ln n - n + O\left( \ln \left( n \right) \right)
n!∼2πn(ne)n n! \sim \sqrt[ ]{ 2\pi n} {\left( \frac{n}{e} \right)}^n

Gamma function

Γ(n)=(n−1)! \Gamma \left( n \right) = \left( n -1 \right) !
Γ(12)=π \Gamma \left( \frac{1}{2} \right) = \sqrt[ ]{ \pi }
Γ(12+n)=(2n)!4nn!π=(2n−1)!!22π=(2n)!22nn!π \Gamma \left( \frac{1}{2} +n \right) = \frac{ \left( 2n \right) !}{ 4^n n! }\sqrt[ ]{ \pi } = \frac{ \left( 2n-1 \right) !! }{ 2^2 } \sqrt[ ]{ \pi } = \frac{\left( 2n \right) !}{2^{2n} n!} \sqrt[ ]{ \pi }