Definition:Exponential Function/Complex/Differential Equation

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Definition

Let $\exp: \C \to \C \setminus \left\{ {0}\right\}$ denote the (complex) exponential function.

The exponential function can be defined as the unique solution $y = f \paren z$ to the first order ODE:

$\dfrac {\d y} {\d z} = y$

satisfying the initial condition $f \paren 0 = 1$.

That is, the defining property of $\exp$ is that it is its own derivative.


The complex number $\exp z$ is called the exponential of $z$.


Also see