Definition:Exponential Function/Real/Power Series Expansion

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