SearchOnMath

Factorials apply only to nonnegative integers but can be extended to complex numbers with positive real parts using the Gamma function, \(\Gamma(n) = \int_0^\infty e^{-x}x^{n-1}dx\).

The heat equation, \(\frac{\partial \varnothing}{\partial t} = D \nabla^2 \varnothing\), uses the Laplacian of \(\varnothing\) to express diffusion through a medium.

Euler's identity, \(e^{i\pi}+1=0\), relates the imaginary unit and two famous irrational numbers in a way that is so simple it has been called beautiful.