444 INTEGRALTRANSFORMSTABLE 11.4 Main properties of the Fourier transform No.. Asymmetric form of the Fourier transform.. Sometimes the alternative Fourier transform is used and called m
Trang 1444 INTEGRALTRANSFORMS
TABLE 11.4 Main properties of the Fourier transform
No Function Fourier transform Operation
1 af1(x) + bf2(x) a 2 f1(u) + b 2 f2(u) Linearity
3 x n (x); n =1, 2, i n 2f(n)
u (u) of the transformDifferentiation
5 f x(n) (x) (iu) n f2(u) Differentiation
6
∞
–∞
f1(ξ)f2(x – ξ) dξ f21(u)2 f2(u) Convolution
For brevity, we rewrite formula (11.4.1.2) as follows:
f (x) =F– 1{f2(u)} or f (x) =F– 1{f(u), x2 }
11.4.1-2 Asymmetric form of the Fourier transform Alternative Fourier transform
1◦ Sometimes it is more convenient to define the Fourier transform by
ˇ
f (u) =
∞
–∞ f (x)e
–iux dx.
In this case, the Fourier inversion formula reads
f (x) = 1
2π
∞
–∞
ˇ
f (u)e iux du.
2◦ Sometimes the alternative Fourier transform is used (and called merely the Fourier
transform), which corresponds to the renaming e–iux e iux on the right-hand sides of
(11.4.1.1) and (11.4.1.2)
11.4.1-3 Convolution theorem Main properties of the Fourier transforms
1◦ The convolution of two functions f (x) and g(x) is defined as
f(x) ∗ g(x)≡ √1
2π
∞
–∞ f (x – t)g(t) dt.
By performing substitution x – t = u, we see that the convolution is symmetric with respect
to the convolved functions: f (x) ∗ g(x) = g(x) ∗ f(x).
The convolution theorem states that
F5f(x) ∗ g(x)6=F5f (x)6
F5g(x)6
2◦ The main properties of the correspondence between functions and their Fourier
trans-forms are gathered in Table 11.4
Trang 211.4.1-4 n-dimensional Fourier transform.
The Fourier transform admits n-dimensional generalization:
2
(2π) n/2
Rn f (x)e–i(u⋅x)dx, (u⋅x) = u1x1+· · · + u n x n, (11.4.1.3)
where f (x) = f (x1, , x n), 2f (u) = f (u1, , u n ), and dx = dx1 dx n.
The corresponding inversion formula is
(2π) n/2
Rn
2
f (u)e i(u⋅x)du, d u = du1 du n.
The Fourier transform (11.4.1.3) is frequently used in the theory of linear partial
differ-ential equations with constant coefficients (xRn).
11.4.2 Fourier Cosine and Sine Transforms
11.4.2-1 Fourier cosine transform
1◦ Let a function f (x) be integrable on the semiaxis 0 ≤ x < ∞ The Fourier cosine transform is defined by
2
fc(u) = 2
π
∞
0 f (x) cos(xu) dx, 0< u < ∞. (11.4.2.1) For given 2fc(u), the function can be found by means of the Fourier cosine inversion formula
π
∞
0
2
fc(u) cos(xu) du, 0< x < ∞. (11.4.2.2) The Fourier cosine transform (11.4.2.1) is denoted for brevity by 2fc(u) =Fc
5
f (x)6
2◦ It follows from formula (11.4.2.2) that the Fourier cosine transform has the property
F2c =1
Some other properties of the Fourier cosine transform:
Fc
5
x2n f (x)6
= (–1)n d2n
du2nFc
5
f (x)6
, n=1, 2, ;
Fc
5
f (x)6
= –u2Fc
5
f (x)6
Here, f (x) is assumed to vanish sufficiently rapidly (exponentially) as x → ∞ For the second formula, the condition f (0) =0is assumed to hold
Parseval’s relation for the Fourier cosine transform:
∞
0 Fc
5
f (x)6
Fc
5
g(x)6
du=
∞
0 f (x)g(x) dx.
There are tables of the Fourier cosine transform (see Section T3.3 and the references listed at the end of the current chapter)
3◦ Sometimes the asymmetric form of the Fourier cosine transform is applied, which is
given by the pair of formulas
ˇ
fc(u) =
∞
0 f (x) cos(xu) dx, f (x) =
2
π
∞
0 ˇ
fc(u) cos(xu) du.
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11.4.2-2 Fourier sine transform
1◦ Let a function f (x) be integrable on the semiaxis0 ≤x<∞ The Fourier sine transform
is defined by
2
fs(u) = 2
π
∞
0 f (x) sin(xu) dx, 0< u < ∞. (11.4.2.3) For given 2fs(u), the function f (x) can be found by means of the inverse Fourier sine transform
f (x) = 2
π
∞
0
2
fs(u) sin(xu) du, 0< x < ∞. (11.4.2.4) The Fourier sine transform (11.4.2.3) is briefly denoted by 2fs(u) =Fs
5
f(x)6
2◦ It follows from formula (11.4.2.4) that the Fourier sine transform has the property
F2s =1
Some other properties of the Fourier sine transform:
Fs
5
x2n f (x)6
= (–1)n d2n
du2nFs
5
f (x)6
, n=1, 2, ;
Fs
5
f (x)6
= –u2Fs
5
f (x)6
Here, f (x) is assumed to vanish sufficiently rapidly (exponentially) as x → ∞ For the second formula, the condition f (0) =0is assumed to hold
Parseval’s relation for the Fourier sine transform:
∞
0 Fs
5
f (x)6
Fs
5
g(x)6
du=
∞
0 f (x)g(x) dx.
There are tables of the Fourier cosine transform (see Section T3.4 and the references listed at the end of the current chapter)
3◦ Sometimes it is more convenient to apply the asymmetric form of the Fourier sine
transform defined by the following two formulas:
ˇ
fs(u) =
∞
0 f (x) sin(xu) dx, f (x) =
2
π
∞
0
ˇ
fs(u) sin(xu) du.
11.5 Other Integral Transforms
11.5.1 Integral Transforms Whose Kernels Contain Bessel
Functions and Modified Bessel Functions
11.5.1-1 Hankel transform
1◦ The Hankel transform is defined as follows:
2
f ν (u) =
∞
0 xJ ν (ux)f (x) dx, 0< u < ∞, (11.5.1.1)
Trang 4where ν > –12 and J ν (x) is the Bessel function of the first kind of order ν (see Section SF.6).
For given 2f ν (u), the function f (x) can be found by means of the Hankel inversion
formula
f (x) =
∞
0 uJ ν (ux) 2 f ν (u) du, 0< x < ∞. (11.5.1.2)
Note that if f (x) = O(x α ) as x →0, where α + ν +2>0, and f (x) = O(x β ) as x → ∞, where β + 32 <0, then the integral (11.5.1.1) is convergent
The inversion formula (11.5.1.2) holds for continuous functions If f (x) has a (finite) jump discontinuity at a point x = x0, then the left-hand side of (11.5.1.2) is equal to 1
2[f (x0–0) + f (x0+0)] at this point
For brevity, we denote the Hankel transform (11.5.1.1) by 2f ν (u) = H ν5f (x)6
2◦ It follows from formula (11.5.1.2) that the Hankel transform has the propertyH2
ν =1 Other properties of the Hankel transform:
H ν
1
x f (x)
2ν H ν–15
f (x)6
2ν H ν+15
f (x)6
,
H ν5f (x)6
= (ν –1)u
2ν H ν+15
f (x)6
– (ν +1)u
2ν H ν–15
f (x)6
,
H ν
f (x) + 1
x f
(x) – ν2
x2f (x)
= –u2H ν5f (x)6
The conditions
lim
x→0
x ν f (x)
=0, lim
x→0
x ν+1f (x)
=0, lim
x→∞
x1 2f (x)
=0, lim
x→∞
x1 2f (x)
=0
are assumed to hold for the last formula
Parseval’s relation for the Hankel transform:
∞
0 uH ν5f (x)6
H ν5g(x)6
du=
∞
0 xf (x)g(x) dx, ν > –
1
2.
11.5.1-2 Meijer transform
The Meijer transform is defined as follows:
ˆ
f μ (s) = 2
π
∞
0
√
sx K μ (sx)f (x) dx, 0< s < ∞,
where K μ (x) is the modified Bessel function of the second kind (the Macdonald function)
of order μ (see Section SF.7).
For given 2f μ (s), the function f (x) can be found by means of the Meijer inversion
formula
f (x) = 1
i √
2π
c+i∞
c–i∞
√
sx I μ (sx) ˆ f μ (s) ds, 0< x < ∞, where I μ (x) is the modified Bessel function of the first kind of order μ (see Section SF.7).
For the Meijer transform, a convolution is defined and an operational calculus is developed
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11.5.1-3 Kontorovich–Lebedev transform
The Kontorovich–Lebedev transform is introduced as follows:
F(τ ) =
∞
0 K iτ (x)f (x) dx, 0< τ < ∞, where K μ (x) is the modified Bessel function of the second kind (the Macdonald function)
of order μ (see Section SF.7) and i = √
–1
For given F (τ ), the function can be found by means of the Kontorovich–Lebedev inversion formula
f (x) = 2
π2x
∞
0 τ sinh(πτ )K iτ (x)F (τ ) dτ , 0< x < ∞.
11.5.1-4 Y -transform.
The Y -transform is defined by
F ν (u) =
∞
0
√
ux Y ν (ux)f (x) dx,
where Y ν (x) is the Bessel function of the second kind of order ν.
Given a transform F ν (u), the inverse Y -transform f (x) is found by the inversion formula
f (x) =
∞
0
√
uxHν (ux)F ν (u) du,
where Hν (x) is the Struve function, which is defined as
Hν (x) =
∞
j=0
(–1) (x/2)ν+2j+1
Γ j+ 32
Γ ν + j + 32
11.5.2 Summary Table of Integral Transforms Areas of Application
of Integral Transforms
11.5.2-1 Summary table of integral transforms
Table 11.5 summarizes the integral transforms considered above and also lists some other integral transforms; for the constraints imposed on the functions and parameters occurring
in the integrand, see the references given at the end of this section
11.5.2-2 Areas of application of integral transforms
Integral transforms are widely used for the evaluation of integrals, summation of series, and solution of various mathematical equations and problems In particular, the application
of an appropriate integral transform to linear ordinary differential, integral, and difference equations reduces the problem to a linear algebraic equation for the transform; and linear partial differential equations are reduced to an ordinary differential equation
Trang 6TABLE 11.5 Summary table of integral transforms Integral transform Definition Inversion formula
Laplace
transform f2(p) =∞
0 e –px f (x) dx f (x) = 1
2πi
c+i∞
c–i∞ e
px f2(p) dp
Laplace–
Carlson
transform
2
f (p) = p
0 e –px f (x) dx f (x) = 1
2πi
c+i∞
c–i∞ e
px f2(p)
p dp
Two-sided
Laplace
transform
2
f ∗ (p) =
–∞ e
–px f (x) dx f (x) = 1
2πi
c+i∞
c–i∞ e
px f2∗ (p) dp
Fourier
transform f2(u) = √1
2π
–∞ e
–iux f (x) dx f (x) = √1
2π
–∞ e iux f2(u) du
Fourier sine
transform f2s(u) = 2
π
0 sin(xu)f (x) dx f (x) = 2
π
0 sin(xu) 2 fs(u) du Fourier cosine
transform f2c(u) = 2
π
0 cos(xu)f (x) dx f (x) = 2
π
0 cos(xu) 2 fc(u) du Hartley
transform f2h(u) =√1
2π
–∞ (cos xu + sin xu)f (x) dx f (x) = √1
2π
–∞ (cos xu + sin xu) 2 fh(u) du Mellin
transform f$(s) = ∞
0 x
s–1f (x) dx f (x) = 1
2πi
c+i∞
c–i∞ x
–s f$(s) ds
Hankel
transform f$ν (w) = ∞
0 xJ ν (xw)f (x) dx f (x) =
0 wJ ν (xw) $ f ν (w) dw
Y-transform F ν (u) =
0
√
ux Y ν (ux)f (x) dx f (x) =
0
√
uxHν (ux)F ν (u) du
Meijer
transform
(K-transform)
$
f (s) = 2
π
0
√
sx K ν (sx)f (x) dx f (x) = 1
i √
2π
c+i∞
c–i∞
√
sx I ν (sx) $ f (s) ds
Bochner
transform
2
f (r) =
0 J n/2–1(2πxr )G(x, r)f (x) dx,
G (x, r) =2πr (x/r) n/2, n=1, 2, f (x) =
0 J n/2–1(2πrx )G(r, x) 2 f (r) dr
Weber
transform
F a (u) =
a W ν (xu, au)xf (x) dx,
W ν (β, μ)≡J ν (β)Y ν (μ) – J ν (μ)Y ν (β)
f (x) =
0
W ν (xu, au)
J ν2(au) + Y ν2(au) uF a (u) du
Hardy
transform
F (u) =
0 C ν (xu)xf (x) dx,
C ν (z)≡ cos(πp)J ν (z) + sin(πp)Y ν (z)
f (x) =
0 Φ(xu)uF (u) du Φ(z)=∞
n=0
(– 1 )n(z/2 )ν+2p+2n Γ(p+n+1 )Γ(ν+p+n+1 )
Kontorovich–
Lebedev
transform
F (τ ) =
0 K iτ (x)f (x) dx f (x) = 2
π2x
0 τ sinh(πτ )K iτ (x)F (τ ) dτ
Meler–Fock
transform F2(τ ) =
1 P–1+iτ (x)f (x) dx f (x) =
0 τ tanh(πτ )P–1+iτ (x) 2 F (τ ) dτ
Euler
transform of
the 1st kind*
F (x) = 1
Γ(μ)
x
a
f (t) dt (x – t)1–μ
0< μ <1, x > a f (x) =
1
Γ( 1– μ)
d dx
x
a
F (t) dt (x – t) μ
Trang 7450 INTEGRALTRANSFORMS
TABLE 11.5 (continued)
Summary table of integral transforms Integral transform Definition Inversion formula
Euler
transform of
the 2nd kind*
F (x) = 1
Γ(μ)
a
x
f (t) dt (t – x)1–μ
0< μ <1, x < a f (x) = –
1
Γ( 1– μ)
d dx
a
x
F (t) dt (t – x) μ
Gauss
transform** F (x) =
1
√ πa
–∞exp
–(x – t) 2
a
f (t) dt f (x) = exp
–a 4
d2
dx2
F (x)
Hilbert
transform*** F$(s) =1
π
–∞
f (x)
x – s dx f (x) = –
1
π
–∞
$
F (s)
s – x ds
NOTATION: i =√
–1, Jμ (x) and Y μ (x) are the Bessel functions of the first and the second kind, respectively;
I μ (x) and K μ (x) are the modified Bessel functions of the first and the second kind, respectively; P μ (x) is the
Legendre spherical function of the second kind; and Hμ (x) is the Struve function (see Subsection 11.5.1-4).
REMARKS.
* The Euler transform of the first kind is also known as Riemann–Liouville integral (the left fractional
integral of order μ or, for short, the fractional integral) The Euler transform of the second kind is also called the right fractional integral of order μ.
** If a =4, the Gauss transform is called the Weierstrass transform In the inversion formula, the exponential
is represented by an operator series: exp
k dx d22
≡1 + ∞
n=1
k n
n! d
2n
dx2n.
*** In the direct and inverse Hilbert transforms, the integrals are understood in the sense of the Cauchy principal value.
Table 11.6 presents various areas of application of integral transforms with literature references
Example.
Consider the Cauchy problem for the integro-differential equation
dy
dx +
x
0 K (x – t)y(t) dt = f (x) (0 ≤t<∞) (11.5.2.1) with the initial condition
Multiply equation (11.5.2.1) by e–px and then integrate with respect to x from zero to infinity Using
properties 7 and 12 of the Laplace transform (Table 11.1) and taking into account the initial condition (11.5.2.2),
we obtain a linear algebraic equation for the transform2y(p):
p 2y(p) – a + 2 K (p) 2y(p) = 2 f (p).
It follows that
2y(p) = f2(p) + a
p+ 2K (p).
By the inversion formula (11.2.1.2), the solution to the original problem (11.5.2.1)–(11.5.2.2) is found in the form
y (x) = 1
2πi
c+i∞
c–i∞
2
f (p) + a
p+ 2K (p) e
px dp, i2= –1 (11.5.2.3)
Consider the special case of a =0 and K(x) = cos(bx) From row 10 of Table 11.2 it follows that
2
K (p) = p
p2+ b2 Rearrange the integrand of (11.5.2.3):
2
f (p)
p+ 2K (p) =
p2+ b2
p (p2+ b2+ 1)f2(p) =
1
p (p2+ b2+ 1)
2
f (p).
In order to invert this expression, let us use the convolution theorem (see formula 16 of Subsection T3.2.1) as well as formulas 1 and 28 for the inversion of rational functions, Subsection T3.2.1 As a result, we arrive at the solution in the form
y (x) =
x
0
b2+ cos t √
b2+ 1
b2+ 1 f (x – t) dt.