# Natural Exponential Function

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**Natural Exponential Function** is a exponential function whose exponent base is [math]\displaystyle{ e }[/math].

**Context:**- It can be defined by the following power series: [math]\displaystyle{ e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{z^n}{n!}. }[/math]
- It can be defined as the inverse transformation of the natural logarithm.
- It can be represented as symbol
*exp*instead of the conventionally*e*, i.e. [math]\displaystyle{ e^x=exp(x) }[/math] - In the context of Differential Equations, it appears in the general solution of the first-order differential equation [math]\displaystyle{ y(t)\propto exp(-\int a(t) dt) }[/math] and as complex-valued solution in the solution of a second-order linear differential equation as [math]\displaystyle{ e^{2it}=cos2t+sin2t }[/math]

**Example(s):****Counter-Example(s):****See:**Complex Exponential Function, Exponential Function, Natural Logarithm.

## References

- http://en.wikipedia.org/wiki/Exponential_function
- (Martin Braun, 1992), Differential Equations and their Applications [1]