Above, we defined as the particular real number satisfying
which gave us
when . From this expression,
we have, as
,
or
This is one way to define . Another way to arrive at the same
definition is to ask what logarithmic base gives that the derivative of
is . We denote by .
Numerically, is a transcendental number (a type of irrational
number3.5), so its decimal expansion never repeats.
The initial decimal expansion of is given by3.6
Any number of digits can be computed from the formula
by making sufficiently small.
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Detailed derivation of the Discrete Fourier Transform (DFT) and its associated mathematics, including elementary audio signal processing applications and matlab programming examples.