SearchOnMath

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

Prime numbers are key to Euler's product formula, \(\sum_{k=1}^{\infty}\frac{1}{k^s}=\prod_{p \;\; \mathrm{prime}}\frac{1}{1-p^{-s}}\), which establishes equality between an infinite sum and product for fixed \(s>1\).

The wave equation, \(\frac{\partial^2u}{\partial t^2}= c^2\nabla^2 u\), relates how \(u\) varies with time and with the spatial coordinates as a wave propagates with speed \(c\).