natural exponential function

Key facts

General definition

exp ⁡ z = e z {\displaystyle \exp z=e^{z}}

Domain

C {\displaystyle \mathbb {C} }

Image

{ ( 0 , ∞ ) for z ∈ R C ∖ { 0 } for z ∈ C {\displaystyle {\begin{cases}(0,\infty )&{\text{for }}z\in \mathbb {R} \\\mathbb {C} \setminus \{0\}&{\text{for }}z\in \mathbb {C} \end{cases}}}

Value at 1

e

Fixed point

− W n (−1) for n ∈ Z {\displaystyle n\in \mathbb {Z} }

Reciprocal

exp ⁡ ( − z ) {\displaystyle \exp(-z)}

Inverse

Natural logarithm , Complex logarithm

Derivative

d d z exp ⁡ z = exp ⁡ z {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \!\,z}}\exp z=\exp z}

Antiderivative

∫ exp ⁡ z d z = exp ⁡ z + C {\displaystyle \int \exp z\,dz=\exp z+C}

Taylor series

exp ⁡ z = ∑ n = 0 ∞ z n n ! {\displaystyle \exp z=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}

via Wikipedia infobox