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Concept
Exponential function
Summary
The exponential function is a mathematical function denoted by or (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
which, along with the definition , shows that for positive integers n, and relates the exponential function to the elementary notion of exponentiation. The base of the exponential function, its value at 1, , is a ubiquitous mathematical constant called Euler's number.
While other continuous nonzero functions that satisfy the exponentiation identity are also known as exponential functions, the exponential function exp is the unique real-valued function of a real variable whose derivative is itself and whose value at 0 is 1; that is, for all real x, and Thus, exp is sometimes called the natural exponential function to distinguish it from these other exponential functions, which are the functions of the form , where the base b is a positive real number. The relation for positive b and real or complex x establishes a strong relationship between these functions, which explains this ambiguous terminology.
The real exponential function can also be defined as a power series. This power series definition is readily extended to complex arguments to allow the complex exponential function to be defined. The complex exponential function takes on all complex values except for 0 and is closely related to the complex trigonometric functions, as shown by Euler's formula.
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