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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 \(c\).

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 natural logarithm can be expanded into a series, yielding \(\ln(1+x) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}x^k}{k}\).