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The Taylor series for an infinitely differentiable \(f\) at point \(a\), given by \(f(x) = \sum_{k=0} ^ {\infty} \frac {f^{(k)}(a)}{k!} \, (x-a)^{k}\), allows \(f\) to be expressed in terms of its derivatives.

A definition of the exponential function that has found application in numerous approximation schemes is \(e^x = \lim_{y\to\infty} \left(1+\frac{x}{y}\right)^y\).

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.