- 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!}}}