§1.14 Integral Transforms
Contents
- §1.14(i) Fourier Transform
- §1.14(ii) Fourier Cosine and Sine Transforms
- §1.14(iii) Laplace Transform
- §1.14(iv) Mellin Transform
- §1.14(v) Hilbert Transform
- §1.14(vi) Stieltjes Transform
- §1.14(vii) Tables
- §1.14(viii) Compendia
§1.14(i) Fourier Transform
The Fourier transform of a real- or complex-valued function
is
defined by
(Some references replace
by
.)
If
is absolutely integrable on
, then
is
continuous,
as
, and
¶ Inversion
Suppose that
is absolutely integrable on
and of
bounded variation in a neighborhood of
(§1.4(v)). Then
where the last integral denotes the Cauchy principal value (1.4.25).
In many applications
is absolutely integrable and
is continuous
on
. Then
¶ Convolution
For Fourier transforms, the convolution
of two functions
and
defined on
is given by
If
and
are absolutely integrable on
, then so
is
, and its Fourier transform is
, where
is the
Fourier transform of
.
¶ Parseval’s Formula
Suppose
and
are absolutely integrable on
, and
and
are their respective Fourier transforms. Then
(1.14.8) is Parseval’s formula.
¶ Uniqueness
If
and
are continuous and absolutely integrable on
, and
for all
, then
for all
.
§1.14(ii) Fourier Cosine and Sine Transforms
These are defined respectively by
¶ Inversion
If
is absolutely integrable on
and of bounded variation
(§1.4(v)) in a neighborhood of
, then
¶ Parseval’s Formula
If
,
is of bounded variation on
and
as
, then
where
and
are respectively the cosine and sine transforms of
.
§1.14(iii) Laplace Transform
Suppose
is a real- or complex-valued function and
is a real or
complex parameter. The Laplace transform of
is defined by
Alternative notations are
,
, or even
, when it is not important to display all the variables.
¶ Convergence and Analyticity
Assume that on
is piecewise continuous and of
exponential growth,
that is, constants
and
exist such that
Then
is an analytic function of
for
. Moreover,
Throughout the remainder of this subsection we assume (1.14.18) is
satisfied and
.
¶ Inversion
If
is continuous and
is piecewise continuous on
,
then
Moreover, if
in some half-plane
and
, then (1.14.20) holds for
.
¶ Translation
¶ Differentiation and Integration
If
is piecewise continuous, then
If also
exists, then
¶ Periodic Functions
If
and
for
, then
Alternatively if
for
, then
¶ Derivatives
If
is continuous on
and
is piecewise continuous on
, then
If
and
are piecewise continuous on
with
discontinuities at (
)
, then
Next, assume
,
,
,
are continuous and each
satisfies (1.14.18). Also assume that
is piecewise
continuous on
. Then
¶ Convolution
For Laplace transforms, the convolution of two functions
and
, defined on
, is
If
and
are piecewise continuous, then
¶ Uniqueness
If
and
are continuous and
,
then
.
§1.14(iv) Mellin Transform
The Mellin transform of a real- or complex-valued function
is
defined by
Alternative notations for
are
and
.
If
is integrable on
for all
in
, then the integral (1.14.32) converges and
is an analytic function of
in the vertical strip
. Moreover, for
,
Note: If
is continuous and
and
are real numbers such
that
as
and
as
, then
is integrable on
for all
.
¶ Inversion
Suppose the integral (1.14.32) is absolutely convergent on the
line
and
is of bounded variation in a
neighborhood of
. Then
If
is continuous on
and
is integrable on
, then
¶ Parseval-type Formulas
Suppose
and
are absolutely integrable on
and either
or
is absolutely integrable on
.
Then for
,
When
is real and
,
¶ Convolution
Let
If
and
are absolutely integrable on
, then for
,
§1.14(v) Hilbert Transform
The Hilbert transform of a real-valued function
is
defined in the following equivalent ways:
¶ Inversion
Suppose
is continuously differentiable on
and
vanishes outside a bounded interval. Then
¶ Inequalities
If
,
, is integrable on
, then so is
and
where
when
, or
when
. These bounds are sharp, and
equality holds when
.
§1.14(vi) Stieltjes Transform
The Stieltjes transform of a real-valued function
is defined by
Sufficient conditions for the integral to converge are that
is a positive
real number, and
as
, where
.
If the integral converges, then it converges uniformly in any compact domain in
the complex
-plane not containing any point of the interval
.
In this case,
represents an analytic function in the
-plane cut along the negative real axis, and
¶ Inversion
If
is absolutely integrable on
for every finite
, and the
integral (1.14.47) converges, then
for all values of the positive constant
for which the right-hand side
exists.
¶ Laplace Transform
If
is piecewise continuous on
and the integral
(1.14.47) converges, then
§1.14(vii) Tables
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§1.14(viii) Compendia
For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).



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