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Another common definition of the exponential function to the natural basis, \(e^x\), comes from writing it as the series \(e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}\).

The Schrödinger equation, \(i \hbar \frac{\partial \Psi}{\partial t} = \hat H \Psi\), is a partial differential equation for the wave function \(\Psi\) of quantum mechanics.

Complicated convolution operations become simple multiplications via the Laplace transform \(F(s) = \int_{-\infty}^{\infty}f(x)e^{-s x}dx\) for function \(f\).