Inverse Laplace transform.Given the transform 2f p, the function f x can be found by means of the inverse Laplace transform f x = 1 2πi c+i∞ c–i∞ 2 f pe px dp, i2 = –1, 11.2.1.2 where t
Trang 111.2.1-2 Inverse Laplace transform.
Given the transform 2f (p), the function f (x) can be found by means of the inverse Laplace
transform
f (x) = 1
2πi
c+i∞
c–i∞
2
f (p)e px dp, i2 = –1, (11.2.1.2) where the integration path is parallel to the imaginary axis and lies to the right of all singularities of 2f (p), which corresponds to c > σ0
The integral in inversion formula (11.2.1.2) is understood in the sense of the Cauchy principal value: c+i∞
c–i∞
2
f (p)e px dp= lim
ω→∞
c+iω
c–iω
2
f (p)e px dp
In the domain x <0, formula (11.2.1.2) gives f (x)≡ 0
Formula (11.2.1.2) holds for continuous functions If f (x) has a (finite) jump discontinu-ity at a point x = x0>0, then the left-hand side of (11.2.1.2) is equal to 12[f (x0–0)+f (x0+0)]
at this point (for x0 =0, the first term in the square brackets must be omitted)
For brevity, we write the Laplace inversion formula (11.2.1.2) as follows:
f (x) =L– 15 2f (p)6
or f (x) =L– 15 2f (p), x6
There are tables of direct and inverse Laplace transforms (see Sections T3.1 and T3.2), which are handy in solving linear differential and integral equations
11.2.2 Main Properties of the Laplace Transform Inversion
Formulas for Some Functions
11.2.2-1 Convolution theorem Main properties of the Laplace transform
1◦ The convolution of two functions f (x) and g(x) is defined as an integral of the form
x
0 f (t)g(x – t) dt, and is usually denoted by f (x) ∗ g(x) or
f (x) ∗ g(x) =
x
0 f (t) g(x – 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
L5f (x) ∗ g(x)6=L5f (x)6
L5g (x)6 and is frequently applied to solve Volterra equations with kernels depending on the difference
of the arguments
2◦ The main properties of the correspondence between functions and their Laplace
trans-forms are gathered in Table 11.1
3◦ The Laplace transforms of some functions are listed in Table 11.2; for more detailed
tables, see Section T3.1 and the list of references at the end of this chapter
Trang 2438 INTEGRALTRANSFORMS
TABLE 11.1 Main properties of the Laplace transform
1 af1(x) + bf2(x) a 2 f1(p) + b 2 f2(p) Linearity
3 f (ξ) f≡ 0(x – a), for ξ <0 e–ap f2(p) Shift of
the argument
4 x n (x); n =1 , 2, (– 1 )n 2f(n)
p (p) of the transformDifferentiation
x f (x)
∞
p
2
of the transform
8 f x(n) (x) p f2(p) –n
k=1p
n–k f(k–1 )
x (+ 0 ) Differentiation
9 x m f x(n) (x), m =1 , 2, (–1 )m d
m
dp m
*
p f2(p) –n
k=1p
n–k f(k–1 )
x (+ 0 ) +
Differentiation
dx n
x m f (x)
, m≥n (– 1 )m p d
m
dp m f2(p) Differentiation
0 f1(t)f2(x – t) dt f21(p)2 f2(p) Convolution
TABLE 11.2 The Laplace transforms of some functions
No. Function, f (x) Laplace transform, 2f (p) Remarks
5 x a –bx Γ(a +1)(p + b)–a–1 a> – 1
p2– a2
p2– a2
p (ln p + C) is the Euler constantC =0.5772 .
p2+ a2
p2+ a2
a
2√ x
pexp –a √
p
a≥ 0
p2+ a2 J0(x) is the Bessel function
Trang 311.2.2-2 Inverse transforms of rational functions.
Consider the important case in which the transform is a rational function of the form
2
f (p) = R (p)
where Q(p) and R(p) are polynomials in the variable p and the degree of Q(p) exceeds that
of R(p).
Assume that the zeros of the denominator are simple, i.e.,
Q (p)≡const (p – λ1)(p – λ2) (p – λ n)
Then the inverse transform can be determined by the formula
f (x) =
n
k=1
R (λ k)
Q (λ k) exp(λ k x), (11.2.2.2) where the primes denote the derivatives
If Q(p) has multiple zeros, i.e.,
Q (p)≡const (p – λ1)s1(p – λ2)s2 (p – λ m)s m, then
f (x) =
m
k=1
1
(s k–1)! p→slimk
d s k 1
dp s k 1
(p – λ k)s k f2(p)e px
(11.2.2.3)
Example 1 The transform
2
f (p) = b
p2– a2 (a, b real numbers) can be represented as the fraction (11.2.2.1) with R(p) = b and Q(p) = (p – a)(p + a) The denominator Q(p) has two simple roots, λ1= a and λ2= –a Using formula (11.2.2.2) with n =2and Q (p) =2p, we obtain the inverse transform in the form
f (x) = b
2a e
ax– b
2a e
–ax= b
a sinh(ax).
Example 2 The transform
2
f (p) = b
p2+ a2 (a, b real numbers) can be written as the fraction (11.2.2.1) with R(p) = b and Q(p) = (p – ia)(p + ia), i2= – 1 The denominator
Q (p) has two simple pure imaginary roots, λ1= ia and λ2= –ia Using formula (11.2.2.2) with n =2 , we find the inverse transform:
f (x) = b
2ia e
iax– b
2ia e
–iax= –bi
2a
cos(ax) + i sin(ax)
+ bi
2a
cos(ax) – i sin(ax)
= b
a sin(ax).
Example 3 The transform
2
f (p) = ap–n,
where n is a positive integer, can be written as the fraction (11.2.2.1) with R(p) = a and Q(p) = p n The
denominator Q(p) has one root of multiplicity n, λ1 = 0 By formula (11.2.2.3) with m =1and s1= n, we
find the inverse transform:
f (x) = a (n –1 )!x
n–1
Fairly detailed tables of inverse Laplace transforms can be found in Section T3.2
Trang 4440 INTEGRALTRANSFORMS
11.2.2-3 Inversion of functions with finitely many singular points
If the function 2f (p) has finitely many singular points, p1, p2, , p n, and tends to zero
as p → ∞, then the integral in the Laplace inversion formula (11.2.1.2) may be evaluated
using the residue theory by applying the Jordan lemma (see Subsection 11.1.2) In this case
f (x) =
n
k=1
res
p=p k[ 2f (p)e px] (11.2.2.4)
Formula (11.2.2.4) can be extended to the case where 2f (p) has infinitely many singular points In this case, f (x) is represented as an infinite series.
11.2.3 Limit Theorems Representation of Inverse Transforms as
Convergent Series and Asymptotic Expansions
11.2.3-1 Limit theorems
THEOREM1 Let0 ≤x<∞ and 2 f (p) =L5f (x)6
be the Laplace transform of f (x) If
a limit of f (x) as x →0exists, then
lim
x→0f (x) = lim p→∞
p 2 f (p)
THEOREM2 If a limit of f (x) as x → ∞ exists, then
lim
x→∞ f (x) = lim p→0
p 2 f (p)
11.2.3-2 Representation of inverse transforms as convergent series
THEOREM1 Suppose the transform 2f (p)can be expanded into series in negative powers
of p,
2
f (p) =
∞
n=1
a n
p n,
convergent for|p| > R, where R is an arbitrary positive number; note that the transform
tends to zero as|p|→ ∞ Then the inverse transform can be obtained by the formula
f (x) =
∞
n=1
a n (n –1)!x
n–1,
where the series on the right-hand side is convergent for all x.
THEOREM2 Suppose the transform 2f (p), |p| > R, is represented by an absolutely
convergent series,
2
f (p) =
∞
n=0
a n
Trang 5where{λ n}is any positive increasing sequence,0< λ0< λ1<· · · → ∞ Then it is possible
to proceed termwise from series (11.2.3.1) to the following inverse transform series:
f (x) =
∞
n=0
a n Γ(λ n)x
λ n– 1, (11.2.3.2)
whereΓ(λ) is the Gamma function Series (11.2.3.2) is convergent for all real and complex values of x other than zero (if λ0≥ 1, the series is convergent for all x).
11.2.3-3 Representation of inverse transforms as asymptotic expansions as x → ∞.
1◦ Let p = p0be a singular point of the Laplace transform 2f (p) with the greatest real part
(it is assumed there is only one such point) If 2f (p) can be expanded near p = p0 into an absolutely convergent series,
2
f (p) =
∞
n=0
c n (p – p0)λ n (λ0< λ1 <· · · → ∞) (11.2.3.3)
with arbitrary λ n , then the inverse transform f (x) can be expressed in the form of the
asymptotic expansion
f (x) ∼ e p0x∞
n=0
c n Γ(–λn)x
–λ n– 1 as x → ∞. (11.2.3.4)
The terms corresponding to nonnegative integer λ n must be omitted from the summation, sinceΓ(0) =Γ(–1) =Γ(–2) =· · · = ∞.
2◦ If the transform 2f (p) has several singular points, p1, , p m, with the same greatest real part, Re p1 =· · · = Re p m, then expansions of the form (11.2.3.3) should be obtained for each of these points and the resulting expressions must be added together
11.2.3-4 Post–Widder formula
In applications, one can find f (x) if the Laplace transform 2 f (t) on the real semiaxis is known for t = p≥ 0 To this end, one uses the Post–Widder formula
f (x) = lim n→∞
(–1)n
n!
n
x
n+1 2
f(n) t
n
x
(11.2.3.5)
Approximate inversion formulas are obtained by taking sufficiently large positive integer n
in (11.2.3.5) instead of passing to the limit
11.3 Mellin Transform
11.3.1 Mellin Transform and the Inversion Formula
11.3.1-1 Mellin transform
Suppose that a function f (x) is defined for positive x and satisfies the conditions
1
0 |f (x)|x σ1 – 1dx<∞,
∞
1 |f (x)|x σ2 – 1dx<∞
for some real numbers σ1and σ2, σ1< σ2
Trang 6442 INTEGRALTRANSFORMS
The Mellin transform of f (x) is defined by
ˆ
f (s) =
∞
0 f (x)x
s–1dx, (11.3.1.1)
where s = σ + iτ is a complex variable (σ1< σ < σ2)
For brevity, we rewrite formula (11.3.1.1) as follows:
ˆ
f (s) =M{f (x)} or fˆ(s) =M{f (x), s}
11.3.1-2 Inverse Mellin transform
Given ˆf (s), the function f (x) can be found by means of the inverse Mellin transform
f (x) = 1
2πi
σ+i∞
σ–i∞
ˆ
f (s)x–s ds (σ1 < σ < σ2), (11.3.1.2)
where the integration path is parallel to the imaginary axis of the complex plane s and the
integral is understood in the sense of the Cauchy principal value
Formula (11.3.1.2) holds for continuous functions If f (x) has a (finite) jump discontinu-ity at a point x = x0>0, then the left-hand side of (11.3.1.2) is equal to12
f (x0–0)+f (x0+0)
at this point (for x0 =0, the first term in the square brackets must be omitted)
For brevity, we rewrite formula (11.3.1.2) in the form
f (x) =M– 1{fˆ(s)} or f (x) =M– 1{fˆ(s), x}
11.3.2 Main Properties of the Mellin Transform Relation Among the
Mellin, Laplace, and Fourier Transforms
11.3.2-1 Main properties of the Mellin transform
1◦ The main properties of the correspondence between the functions and their Mellin
transforms are gathered in Table 11.3
2◦ The integral relations
∞
0 f (x)g(x) dx =M– 1{fˆ(s)ˆg(1– s)},
∞
0 f (x)g
1
x
dx=M– 1{fˆ(s)ˆg(s)}
hold for fairly general assumptions about the integrability of the functions involved (see Ditkin and Prudnikov, 1965)
11.3.2-2 Relation among the Mellin, Laplace, and Fourier transforms
There are tables of direct and inverse Mellin transforms (see Sections T3.5 and T3.6 and the references listed at the end of the current chapter) that are useful in solving specific integral and differential equations The Mellin transform is related to the Laplace and Fourier transforms as follows:
M{f (x), s}=L{f (e x ), –s}+L{f (e–x ), s}=F{f (e x ), is}, which makes it possible to apply much more common tables of direct and inverse Laplace and Fourier transforms
Trang 7TABLE 11.3 Main properties of the Mellin transform
1 af1(x) + bf2(x) a ˆ f1(s) + b ˆ f2(s) Linearity
of the transform
Squared argument
of the transform
, a >0, β≠ 0 1
β a
s+λ
β fˆ s + λ
β
Power law transform
Γ(s – n) fˆ(s – n) Multiple differentiation
dx
n
f (x) (– 1 )n s n fˆ(s) Multiple differentiation
∞
0 t
β f
1(xt)f2(t) dt fˆ1(s + α) ˆ f2(1– s – α + β) Complicated integration
∞
0
t β f1
x
t
f2(t) dt fˆ1(s + α) ˆ f2(s + α + β +1 ) Complicated integration
11.4 Various Forms of the Fourier Transform
11.4.1 Fourier Transform and the Inverse Fourier Transform
11.4.1-1 Standard form of the Fourier transform
The Fourier transform is defined as follows:
2
f (u) = 1
√
2π
∞
–∞ f (x) e
–iux dx. (11.4.1.1)
For brevity, we rewrite formula (11.4.1.1) as follows:
2
f (u) =F{f (x)} or f2(u) =F{f (x), u}. Given 2f (u), the function f (x) can be found by means of the inverse Fourier transform
f (x) = 1
√
2π
∞
–∞
2
f (u) e iux du (11.4.1.2)
Formula (11.4.1.2) holds for continuous functions If f (x) has a (finite) jump disconti-nuity at a point x = x0, then the left-hand side of (11.4.1.2) is equal to 12
f (x0–0)+f (x0+0)
at this point