Properties of Complex Exponential Function
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Theorem
Let $z \in \C$ be a complex number.
Let $\exp z$ be the exponential of $z$.
Then:
Image of Complex Exponential Function
The image of the complex exponential function is $\C \setminus \left\{ {0}\right\}$.
Exponential of Sum
Let $z_1, z_2 \in \C$ be complex numbers.
Let $\exp z$ be the exponential of $z$.
Then:
- $\map \exp {z_1 + z_2} = \paren {\exp z_1} \paren {\exp z_2}$
Reciprocal of Complex Exponential
- $\dfrac 1 {\map \exp z} = \map \exp {-z}$
Complex Exponential Function has Imaginary Period
- $\map \exp {z + 2 k \pi i} = \map \exp z$
Exponential Function is Continuous
- $\forall z_0 \in \C: \ds \lim_{z \mathop \to z_0} \exp z = \exp z_0$
Derivative of Exponential Function
The complex exponential function is its own derivative.
That is:
- $\map {D_z} {\exp z} = \exp z$