Exponential Function

math

Definition

Exponential Function

An exponential function is a function where the function’s parameters are exponents.

Examples:

• $f(x)=e^x$
• $f(x)=a^x$ with $a\in {\mathbb{R}}^{+}$

Properties

Bijectivity

The exponential function is bijective from $\mathbb{R}$ to ${\mathbb{R}}^{+}$.

Continuity

Continuity of Exponential Functions

Exponential functions of form

$$f(x)=e^x,f(x)=a^x$$

with $e$ being the Euler’s number and $a>0$ are continuous.

Periodicity

$$\forall z\in \mathbb{C}:e^{z+2\pi i}=e^z$$

Euler’s Formula

Definition

Euler's Formula

Euler’s formula, named after Leonhard Euler, establishes a relationship between trigonometric functions and the complex exponentiation function.

$$e^{iz}=\cos z+i\sin zz\in \mathbb{C}$$

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