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The notion of a fractional derivative or integral is
naturally defined in terms of the integration and differentiation
theorems for Laplace/Fourier transforms. Let
denote the
bilateral Laplace transform of
:
where
is a complex variable,
typically denotes
time in seconds, `
' means ``equals by definition,'' and we
assume
and all of its integrals and derivatives are absolutely
integrable and approach zero as
. Then the
differentiation theorem for bilateral Laplace transforms states
that
where
denotes the
th derivative of
with respect
to
.2 The proof is quickly derived
using integration by parts.3
The integration theorem for Laplace transforms follows as a corollary:
The Laplace transform specializes to the Fourier transform along the
axis in the complex plane, where
is
radian frequency (radians per second), while
denotes
frequency in Hz (cycles per second).
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