For example, given “0+i” as input, the function returns “-1+3.23108914886517E-15i” as output.
Given real number input, the function behaves like the exponential function and models exponential growth.
Notation
The complex exponential function often appears as the Latin letter e to some power.
This notation is equivalent to passing the complex number as input to the exponential function.
We can write another version of this formula that better describes what’s happening in terms of the angle .
The right expression can be expanded further using Euler’s Formula.
In Excel, the complex exponential function is equivalent to this formula.
We can draw the returned complex number in the complex plane like this.
In general, the function’s output can be visualized with the 3D plot below.
The plot’s surface is colored using the output’s phase (angle).
Inverse
The inverse of the complex exponential function is the complex natural logarithm.
Sometimes, this can result in a branch cut, where a different-but-equivalent angle is returned instead.
This is discussed in more depth in the complex natural logarithm article.
IMLN Function
The Excel IMLN function returns the natural logarithm of a complex number.
