Concept# Exponential function

Summary

The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics".
The exponential function satisfies the exponentiation identity
e^{x+y} = e^x e^y \text{ for all } x,y\in\mat

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