Geoscience Reference
In-Depth Information
cos ω(t-λ) = cos ωt cos ωλ + sin ωt sin ωλ and so
(
)
=
(
)
coscos+
(
)
sinsin
= [
()cos]cos+ [
()sindλ]sin
Or
(
)
=
(
)
cos+
(
)
sin
=
[()cos+()sin]
(11b)
Where
()=
and
()=
(
)
cos
(
)
sin
Equation (11b) is the Fourier integral: a Riemann sum of integral consisting of the first of the
RHS of the equation, called the Fourier cosine integral and the second part called the Fourier
sine integral. This integral equation will converge to f(t) when f(t) is continuous and
converges even at points of discontinuity just like a Fourier series.
Note that the function is suppose to be defined from -∞ to +∞, but because of the parity of
the function, we only need the function from 0 to
∞
. This also means that if we only are
interested in the range of 0 to
∞
, we can define the function from -
∞
to any where we want,
then we can have either cosine integral or sine integral by extending the function into
negative range either in an even or odd form. Thus Fourier cosine and sine integrals are
equivalent to the half-range expansion of Fourier series.
4.5 Fourier transforms and theorems
From the complex form of Fourier series given in equation (7) and expression for the
coefficients given in equation (8), we make a transition T→
∞
and introduce again
the variable, ω = nπ/T, with
∆
=1
, since ∆n = 1. Hence equations (7) and (8) would be
written as
(
)
=
∑
()
∆
and
(
)
=
()
, where C
T
(ω) = TC
n
/π. If we let
T→
∞
, C
T
(ω)→C(ω), and so
()=
()
(12)
(
)
= ()
(13)
There are other ways of expressing equations (12) and (13) which are the Fourier transforms
of each other. If we let F(ω) = 2πC(-ω), then
()=
()
(14)
(
)
=
()
(15)
Equations (14) and (15) are called the Fourier transform pair. If f(t) satisfies the Dirichlet
conditions and the integral
|
()
|
∞
is finite, then F(ω) exists for all ω and is called the
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