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The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right.
The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one: Termwise multiplication of two copies of these power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments: In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series.
The real and imaginary parts of the above expression in fact correspond to the series expansions of and the unit circle, it is easy to see that, restricted to real arguments, the definitions of sine and cosine given above coincide with their more elementary definitions based on geometric notions.
From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, .
The derivative (rate of change) of the exponential function is the exponential function itself.
The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.