Mathematical Functions

Definitions

The exponential function $\exp z$ is defined as $\exp z = \sum_{k = 0}^{\infty} \frac{z^{k}}{k!} .$

Fundamental Properties

The absolute value and argument of the exponential function satisfy $\begin{aligned} | \exp z | & = \exp Re z \\ \arg \exp z & = Im z + 2 π k, \end{aligned}$ with some $k \in ℤ$ . It is periodic in the imaginary direction: $\exp (z + 2 π i) = \exp z .$

Identities

For complex numbers $a, b, z$ and integer $n$ the following holds: $\begin{aligned} \exp (a + b) & = (\exp a) (\exp b) \\ {(\exp z)}^{n} & = \exp (n z) \\ \exp (i z) & = \cos z + i \sin z . \end{aligned}$

Real value of $\exp (x + i y)$ .

The Exponential ← Function

Definitions

Fundamental Properties

Identities

The Exponential

← Function