Definition:Exponential Function/Real/Power Series Expansion
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Definition
Let $\exp: \R \to \R_{>0}$ denote the (real) exponential function.
The exponential function can be defined as a power series:
- $\exp x := \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$
The number $\exp x$ is called the exponential of $x$.
Exponential Series
The power series expansion of the exponential function:
- $\map \exp z = 1 + \dfrac z {1!} + \dfrac {z^2} {2!} + \dfrac {z^3} {3!} + \cdots + \dfrac {z^n} {n!} + \cdots$
is known as the exponential series.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
- 1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.2$ Series
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $24$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $24$
- Weisstein, Eric W. "Exponential Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialFunction.html