Laplace Transform of Products of Bessel Functions: A
Visitation of Earlier Formulas
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Kausel, Eduardo, and Mirza M. Irfan Baig. Laplace Transform of
Products of Bessel Functions: A Visitation of Earlier Formulas.
Quarterly of Applied Mathematics 70(1): 77–97, 2012.
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http://dx.doi.org/10.1090/s0033-569x-2011-01239-2
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American Mathematical Society (AMS)
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Quarterly of Applied Mathematics, Vol. LXX (1), March 2012, 77-97
QUARTERLY OF APPLIED MATHEMATICS
VOLUME , NUMBER 0
XXXX XXXX, PAGES 000–000
S 0033-569X(XX)0000-0
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS:
A VISITATION OF EARLIER FORMULAS
By
EDUARDO KAUSEL (Professor of Civil and Environmental Engineering,
Massachusetts Institute of Technology, Cambridge MA 02139 )
and
MIRZA M. IRFAN BAIG (Simpson, Gumpertz & Heger Inc., Waltham MA 02453 )
Abstract. This note deals with the Laplace transforms of integrands of the form
x Jα (ax) Jβ (bx), which are found in numerous fields of application. Specifically, we
provide herein both a correction and a supplement to the list of integrals given in 1997
by Hanson and Puja, who in turn extended the formulas of Eason, Noble and Sneddon
of 1956. The paper concludes with an extensive tabulation for particular cases and range
of parameters.
λ
1. Introduction. In a classic 1956 paper, Eason, Noble and Sneddon – henceforth
ENS for short – presented a general methodology for finding integrals involving products
of Bessel functions, and provided a set of closed-form formulas for cases commonly encountered in engineering science and in applied mathematics. Although these integrals
extended considerably the repertoire of exact formulas available in standard tables such
as Oberhettinger’s or Gradshteyn & Ryzhik’s, even to this day programs such as Mathematica, Maple and Matlab’s symbolic tool seem to have remained unaware of the ENS
paper, for they are unable to provide answers to such integrals. Some four decades later,
Hanson and Puja (1997) – henceforth denoted as HP – reconsidered the ENS paper and
not only extended considerably the formulas therein, but by changing the arguments to
the functions, they arrived at alternative forms which allegedly avoided discontinuities at
certain values of the parameters. Unfortunately, once the arguments in the HP formulas
exceed some threshold value, the computations for some of the integrals suffer a complete breakdown and become useless. This led us to investigate both the reasons for the
erroneous results and also seek a corrected set of formulas, which constitutes the subject
of this paper.
To avoid repetitions, the presentation herein will be rather terse, avoiding needless
explanations of well-known facts and/or of details which can be found in the originals of
the papers referred to. Also, to distinguish clearly between the equations and parameters
2010 Mathematics Subject Classification. 44-00, 44A10, 74-00.
E-mail address: kausel@mit.edu
c
⃝XXXX
Brown University
1
2
EDUARDO KAUSEL
AND
MIRZA M. IRFAN BAIG
used by ENS and HP, we have adopted a revised, more general notation. We defer a
description of the problem itself until section 5 so as to summarize first some needed
preliminary definitions and properties.
2. Laplace transform and its properties. The integrals in question are the Laplace
transforms
∫ ∞
λ
λ
Iαβ ≡ Iαβ (a, b, s) =
xλ Jα (ax) Jβ (bx) e−sx dx
(2.1)
0
with
α + β + λ > −1
if s > 0
α + β + 1 > −λ > −1
if s = 0
&
a ̸= b
α + β + 1 > −λ > 0
if s = 0
&
a=b
where the parameters α, β, λ are assumed to be integers and the arguments a, b, s are
real. These integrals satisfy the following symmetries and recursive properties:
{ ( )λ+1
(
)
1
I λ ab , 1, sb
λ
λ
λ
Iαβ (a, b, s) = Iβα (b, a, s)
Iαβ (a, b, s) =
(2.2)
( 1b )λ+1 αβ
(
)
λ
Iαβ
1, ab , as
a
(
)
(
)
λ+1
λ+1
λ+1
λ+1
λ
λ
= 12 a Iα−1,β
= 21 b Iα,β−1
α Iαβ
+ Iα+1,β
β Iαβ
+ Iα,β+1
(2.3)
λ
∂Iαβ
=
∂a
1
2
(
λ+1
λ+1
Iα−1,β
− Iα+1,β
)
λ
∂Iαβ
=
∂b
1
2
(
λ+1
λ+1
Iα,β−1
− Iα,β+1
)
(2.4)
Combination of two of the above yields
λ
∂Iαβ
α λ
λ+1
= Iα−1,β
− Iαβ
∂a
a
α λ
λ+1
= Iαβ − Iα+1,β
a
λ
∂Iαβ
β λ
λ+1
= Iα,β−1
− Iαβ
∂b
b
β λ
λ+1
= Iαβ − Iα,β+1
b
(2.5)
In addition,
λ
Iαβ
=−
λ−1
∂Iαβ
∂s
(2.6)
It can also be shown via integration by parts that
[
]
λ−1
λ
λ
λ
(α + β − λ) Iαβ
= a Iα−1,β
+ b Iα,β−1
− s Iαβ
+ xλ Jα (ax) Jβ (bx) x=0
( λ
)
( λ
)
[
]
λ−1
λ
λ
λ
− Iα−1,β
− Iα,β−1
λ Iαβ
= a 12 Iα+1,β
+ b 12 Iα,β+1
+ s Iαβ
− xλ Jα (ax) Jβ (bx) x=0
(2.7)
provided α + β + λ ≥ 0. The boundary term at x = 0 vanishes if α + β + λ > 0. For
example, if λ = 0, α = β = 1, then
( 0
)
−1
0
0
I11
= 12 a I0,1
+ b I1,0
− s I11
(2.8a)
(
)
(
)
0
0
0
0
0
0 = 21 a I2,1
− I0,1
+ 12 b I1,2
− I1,0
+ s I1,1
(2.8b)
both of which can be useful.
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
3. Definitions, parameters and fundamental relations.
√
√
2
2
A = A (a, b, s) = (a + b) + s2 , B = B (a, b, s) = (a − b) + s2
L1 = L1 (a, b, s) =
1
2
(A − B) ,
L2 = L2 (a, b, s) =
1
2
(A + B)
√
√
√
2 ab
2 k
2 L1 L2
=√
=
κ=
L1 + L2
1+k
2
(a + b) + s2
L1
ab
1 − κ′
L2
= 2 = 1 =
L2
L2
ab
1 + κ′
(3.5)
√
√
√
2 AB
2 κ′
2
k = 1−k =
=
A+B
1 + κ′
4ab
2
(a + b)
, nab =
(3.2)
(3.4)
′
ν=
(3.1)
(3.3)
√
2
√
(a − b) + s2
1−k
B
=√
=
κ′ = 1 − κ2 =
A
1+k
2
(a + b) + s2
k=
3
(3.6)
a L1
b L1
L21
a2
a
L21
b2
b
=
=
=
=
k
,
n
=
=
=k
ba
2
2
2
2
b
L2
b L2
b
a
L2
a L2
a
(3.7)
with
κ2 < ν < 1,
k 2 < nab < 1,
k 2 < nba < 1,
nab nba = k 2
(3.8)
These parameters satisfy the following useful relationships:
√
√
√
sκ
|a
−
b|
(1 − ν) (ν − κ2 )
|a − b|
s
ν − κ2 =
,
1−ν =
,
=
a+b
a+b
ν
a + b L2 (1 + k)
√
2
L1 = 21 (1 − κ′ ) (a + b) + s2
,
1 − κ′ √
=
ab
κ
2
2
(L1 + L2 ) = (a + b) + s2 ,
(
)
L21 + L22 = 21 A2 + B 2 = a2 + b2 + s2 ,
L1 L2 =
1
4
(
)
A2 − B 2 = ab,
√
(3.9)
(1 + κ′ ) (a + b) + s2
(3.10)
1 + κ′ √
=
ab
κ
2
2
(L2 − L1 ) = (a − b) + s2
(3.11)
(
)
κ′
L22 − L21 = AB = L22 1 − k 2 = 4ab 2
κ
(3.12)
(
)
L42 = L22 a2 + b2 + s2 − a2 b2
(3.13)
L2 =
1
2
ka − b =
s2 ka
s2 b
,
=
L21 − a2
b2 − L22
s2
1
=
− 1,
2
2
a − L1
nab
kb − a =
s2 kb
s2 a
= 2
,
2
2
L1 − b
a − L22
b2
s2
1
=
− 1,
2
− L1
nba
2
s2
= 1 − nba (3.14)
L22 − a2
L22
s2
= 1 − nab (3.15)
− b2
4
EDUARDO KAUSEL
AND
MIRZA M. IRFAN BAIG
4. Elliptic integrals. All of the transforms considered in this work result in formulas
involving the complete elliptic integrals of the first, second and third kind. In addition,
ENS introduce a Λ function which coincides with Heuman’s Lambda Naught function
multiplied by 21 π. In the ensuing, we use κ, ν for the ENS arguments and k, n for the HP
arguments to the elliptic integrals. These functions and their mathematical properties
are:
∫ 1
dx
√
√
K (κ) =
2
1 − x 1 − κ2 x2
0
1st kind
(4.1)
∫ π/2
dθ
√
=
0
1 − κ2 sin2 θ
∫ 1√
1 − κ2 x2 dx
√
E (κ) =
1 − x2
0
2nd kind
(4.2)
∫ π/2 √
2
2
1 − κ sin θ dθ
=
0
∫
Π (ν, κ) =
0
∫
=
0
√
1
(1 − ν
π/2
x2 )
√
dx
√
1 − x2 1 − κ2 x2
dθ
(
)√
2
1 − ν sin θ
1 − κ2 sin2 θ
(1 − ν) (ν − κ2 )
√
Π (ν, κ) ≡ π2 Λ0 (ψ\ϑ)
ν
= [E (κ) − K (κ)] F (ψ\ϑ′ ) + K (κ) E (ψ\ϑ′ )
Λ (ν, κ) =
3rd kind
(4.3)
(ENS–3.8)
(4.4)
where Λ0 is Heuman’s function, F and E are the incomplete elliptic integrals of the first
and second kind, respectively, and both ϑ = arcsin κ, ψ are defined by the unnumbered
equations above ENS-3.8
sin2 ψ =
ν − κ2
,
νκ′ 2
κ2 (1 − ν)
νκ′ 2
(4.5)
Πba ≡ Π (nba , k)
(4.6)
cos2 ψ =
We also define
Πab ≡ Π (nab , k) ,
with nab , nba given by 3.7.
Landen transformation:
1 − κ′
1
, K (k) = 12 (1 + κ′ ) K (κ) , E (k) =
[E (κ) + κ′ K (κ)]
1 + κ′
1 + κ′
√
2 k
2
κ=
, K (κ) = (1 + k) K (k) , E (κ) =
E (k) − (1 − k) K (k)
1+k
1+k
In Appendix A we also show that
s
Λ (ν, κ) =
[2Π (nab , k) − K (k)] a < b
L2
s
=
[2Π (nba , k) − K (k)] a > b
L2
k=
(4.7)
(4.8)
(4.9)
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
5
Caveat: Users of advanced software such as Mathematica, Maple and Matlab should
beware the arguments being used by the elliptic functions in these programs, for the
manuals are not crystal clear about the matter. Specifically, in Mathematica the functions EllipticK and EllipticE use m = k 2 as argument, i.e. the so-called parameter,
while the same-named functions in Maple and in Matlab’s symbolic tool use instead the
modulus k. To add to the confusion, Matlab’s intrinsic numerical function ellipke also
uses the parameter m instead of the modulus k and restricts it to be real and less than
unity. In the case of multiple solutions for the Laplace transforms, Matlab often gives
only one of these, and also remains silent about some of the underlying assumptions (e.g.
a < b = 1 and so forth).
−1
5. Description of the problem. Consider the particular case of the transform I11
.
From ENS-4.9, this integral is
[
]
)
s
1
κ ( 2
a2 − b2
−1
I11
= √
E (κ) −
a + b2 + 12 s2 K (κ) +
sgn (a − b) Λ (ν, κ)
2ab
2π ab
π ab κ
 b
a>b
1 a
+
1
a=b
2 a
a<b
b
(5.1a)
which can be written compactly as
]
[
)
s
1
κ ( 2
−1
2
1 2
√
I11 =
E (κ) −
a + b + 2 s K (κ)
2ab
π ab κ
(5.1b)
[
]}
(
)
1 { 2
+
a + b2 + a2 − b2 sgn (a − b) π2 Λ (ν, κ) − 1
4ab
whereas from HP-25, this same integral is
(
)
(
)
]
a
s [ 2
−1
I11
=
+
L2 E (k) − L22 + b2 K (k) − a2 − b2 Π (n, k)
(5.2)
2b πabL2
the characteristic n of which coincides with our nab in eq. 3.7. Observe that HP-25
consists of a single expression for the transform whatever the relative values of a, b, and
also uses different arguments and coefficients for the elliptic integrals, while ENS-4.9
exhibits an intrinsic discontinuity at the transition a = b for all values of s. Regrettably,
although the HP expression is indeed continuous, it is not applicable and fails when
a > b, as will be seen.
Now, HP developed their formulas in the context of a problem in the theory of elasticity
concerning a circular load (i.e. “patch”) applied onto an elastic half-space and thus they
argued for the continuity of the functions on physical grounds. Quoting HP (square
brackets are ours):
It is apparent that when the integral evaluation does not contain Heuman’s
Lambda function, only one expression is needed to evaluate the integral
for all values of the parameters. However, when the integral evaluation
also contains Heuman’s Lambda function, two different expressions are
given for the integral depending on the relative values of [a, b]. Since
6
EDUARDO KAUSEL
AND
MIRZA M. IRFAN BAIG
the elastic field inside the half-space is continuous and has continuous
derivatives, it is troublesome that the expressions are different depending
on being inside or outside the radius of loading. Intuition would lead
one to expect a single expression which is valid at every point in the half
space.
Unfortunately, this appeal to physical continuity did not guarantee that the replacement
formulas proposed by HP would be correct, for they actually provide spurious results
when applied for a > b. As it turns out, two separate formulas are needed for the intervals
a < b and a > b which, although distinct, maintain the continuity of the solution and
its derivative at a = b. In the ensuing, we establish the correct mathematical connection
between the ENS and HP formulas and provide an extended set of formulas which are
free from this problem.
6. Relationship between ENS and HP. HP provide a transformation formula
HP-24 between the complete elliptic integral of the third kind with arguments ν, κ and
the elliptic integrals with arguments n, k. In our current notation and after some simple
transformations, their transformation formula would read
{
}
(1 + k) (a + b) πL2
Π (ν, κ) =
H (a − b) + K (k) − 2Π (n, k)
(6.1)
a−b
s
which can also be written as
sgn (a − b) Λ (ν, κ) = πH (a − b) +
s
[K (k) − 2Π (n, k)]
L2
(6.2)
where H (a − b) is the Heaviside function, and n ≡ nab . They state that the proof of this
formula is contained in an earlier paper of theirs, but we have been unable to locate any
such derivation in said paper. Furthermore, that equation holds no obvious connection
to the Landen-Gauss transformation 163.02 given on page 39 in Byrd and Friedman
(1971), which appears to be restricted to the ”hyperbolic” case n < k 2 and thus excludes
the ”circular” case at hand k 2 < n. For this reason, we provide a new proof of the
HP translation formula in Appendix A, and in the process show that another formula is
needed when a > b. With reference to Appendix A, the actual transformation formulas
are:
s
Λ (ν, κ) =
(2Πab − K (k)) a < b
(6.3a)
L2
Λ (ν, κ) =
s
(2Πba − K (k))
L2
a>b
(6.3b)
As also demonstrated in Appendix A, Πab and Πba satisfy the relationship
Πab + Πba − K =
πL2
2s
(6.4)
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
7
Taking into account this property, the two transformation formulas 6.3a,b can be combined into a single expression by means of Heaviside functions as follows:
s
Λ (ψ, κ) =
{ 2Πab [1 − H (a − b)] + 2Πba H (a − b) − K}
L2
{
[
]
}
s
πL2
(6.5)
=
2Πab [1 − H (a − b)] + 2
− Πab + K H (a − b) − K
L2
2s
s
(K − 2Πab )
= π H (a − b) + sgn (a − b)
L2
so
s
sgn (a − b) Λ (ψ, κ) = π H (a − b) +
(K − 2Πab )
(6.6)
L2
which does coincide with HP’s translation formula 6.2. Although this shows 6.2 to be
technically correct, HP’s integral transform formulas are still incorrect when a > b.
This is because in their final expressions for the integrals, they failed to include the
Heaviside term, which is absent when a < b, and simply assumed that the resulting
formulas for that case would be valid throughout, i.e. they reasoned that the formulas
had to be ”continuous”. In addition, Πab is ill-conditioned when a > b. In retrospect, it
seems peculiar that HP should have argued against the discontinuous Lambda function
in ENS, only to replace it with yet another discontinuous function and then ”forgot” –
or proceeded to deliberately ignore – this very discontinuity.
Fortunately, to circumvent this problem, it suffices to make use of 6.4 and replace Πab
π
in all HP formulas by Πab = K (k) − Πba + 2s
L2 whenever a > b . This is possible
because for all a, b, s > 0, k < 1, nab < 1, nba < 1, in which case 6.4 has no singularities
and is a continuous function of the ratio a/b, even if 6.4 itself remains intrinsically illconditioned because either Πab or Πba attains large values, especially when s is small.
This is because for a > b, nab → 1, and Πab → ∞, and vice-versa for Πba .
To demonstrate the application of the preceding transformation formulas, substitute
6.3a into the ENS form 5.1, and consider also the equivalences 4.16, which yields
{
]
}
[
)
s
1 + κ′
2
κ2 ( 2
−1
2
2
′
I11 = √
E (k) −
κ +
2a + 2b + s
K (k)
κ
κ (1 + κ′ )
4ab
π ab
(6.7)
s a2 − b2
a
+
(K (k) − 2Πab ) +
πL2 2 ab
2b
Using the definitions for k, κ and other parameters in section 3, it can be shown that
]
[
)
[ 2 1( 2
)]
2
κ2 ( 2
1
′
2
2
κ
+
2a
+
2b
+
s
=√
L2 + 2 a + b2
(6.8)
′
κ (1 + κ )
4ab
ab L2
Hence
−1
I11
=
(
)
(
)
]
s [ 2
a
L2 E (k) − L22 + b2 K (k) − a2 − b2 Πab + ,
πab L2
2b
a<b
(6.9)
which coincides with HP-25 (i.e. eq. 5.2). On the other hand, making use of eq. 6.3b
instead, we obtain
(
)
(
)
]
s [ 2
b
−1
a>b
(6.10)
I11
=
L2 E (k) − L22 + a2 K (k) − b2 − a2 Πba + ,
πab L2
2a
8
EDUARDO KAUSEL
AND
MIRZA M. IRFAN BAIG
which agrees perfectly with the formula that would be obtained by simply exchanging
−1
a, b in 6.9, which must hold because of the symmetry of I11
with respect to the Bessel
π
indices. Finally, if we substitute Πab = K (k) − Πba + 2s L2 into eq. 6.9, we once more
recover 6.10. Thus, this shows that all is consistent.
7. Tables of integrals. Although many of the integrals listed in the pages that
follow are directly based on the HP and ENS papers, we have seen fit to simplify these
expressions to the extent possible, using for this purpose the very useful equivalences and
properties given in section 3. Hence, they do not quite look like those in HP. This also
meant that each and every formula had to be carefully checked for errors, including tests
against direct numerical integration. Also, since Laplace transforms constitute improper
integrals, it was necessary for us to supplement our numerical quadrature with formulas
for integrating the tails, a task that is presented in Appendix C.
We have made an utmost effort in avoiding mistakes in both the transcription and
proofreading of the typeset document, and believe the formulas to be free from error.
Still, readers are strongly encouraged to carefully verify their own personal implementation of these formulas, not only to avoid any remaining, hidden errors, but also to avoid
errors that could have crept in during implementation of the formulas from the published
paper.
In addition, some of the integrals could be verified against tabulations such as Oberhettinger’s and against Matlab symbolic tool, and we identify the formulas thus checked.
In addition, we have also used the recurrence relations to verify some (but not all) of the
integrals. The reason is that although the recurrence relations involve differentiations
and can in principle be used to reduce the expressions to known integrals, the process
can be very tedious because of the complexity of the derivatives of the elliptic functions.
Indeed, even when the operations are carried out with a computer, say with Matlab, it
is often difficult to collect and factorize terms in the resulting expressions, and thus, to
reduce the formulas into a recognizable form.
Finally, because many of the formulas are discontinuous when either s → 0 or a = b,
we have seen fit to provide separate tables for these cases. By and large, the tables list
the integrals in descending value of λ and in increasing order of the Bessel functions.
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
Table 1: Integrals for α = 0, a = 0, s > 0. All of the formulas
been obtained with Matlab’s symbolic tool.
√
L2 = b2 + s2
(
)β
∫ ∞
1
b
0
I0β
(0, b, s) =
e−sx Jβ (bx) dx =
,
L2 L2 + s
0
(
)β
∫ ∞
Jβ (bx)
1
b
−1
I0β
(0, b, s) =
e−sx
dx =
,
x
β L2 + s
0
(
)β
∫ ∞
βL2 + s
b
−2
−sx Jβ (bx)
I0β (0, b, s) =
e
dx =
x2
β (β 2 − 1) L2 + s
0
∫ ∞
Jβ (bx)
−3
I0β
(0, b, s) =
e−sx
dx
x3
0
[( 2
) 2
](
)β
β − 1 L2 + 3s (βL2 + s)
b
=
,
β (β 2 − 1) (β 2 − 4)
L2 + s
9
in this section have
β = 0, 1, 2, · · ·
β = 1, 2, 3, · · ·
β = 2, 3, 4, · · ·
β = 3, 4, 5, · · ·
Table 2: Integrals for s = 0, a > 0, b > 0, in dual format.
√
{ a
2 ab
b, a < b
κ=
≤1
k=
b
a+b
a, a > b
∫
0
I00
∞
(a, b, 0) =
0
∫
0
(a, b, 0) =
I01
∞
0
∫
2 1
J0 (ax) J0 (bx) dx =
K (κ) =
πa+b
{
( )
2
a
πb K ( b )
2
b
πa K a

 1
1
1
J0 (ax) J1 (bx) dx = 1b [1 − H (a − b)] =
b 2
0
∞
a<b
a>b
a<b
a=b
a>b
1
J0 (ax) J1 (bx)
dx =
[(a + b) E (κ) − (a − b) K (κ)]
x
πb
0
(
)
{
2
E( ab)
[( b
)
( b )] a < b
=
a
b
a
a>b
π
a − b K a + bE a
{ a
∫ ∞
1
J1 (ax) J1 (bx)
a<b
−1
b
I11
dx =
(a, b, 0) =
b
a≥b
x
2
0
a
[
]
∫ ∞
2
2
a
+
b
a
+
b
0
J1 (ax) J1 (bx) dx =
I11
(a, b, 0) =
K (κ) − E (κ)
πab (a + b)2
0
( )]
 1 [ (a)
K b − E ab
a<b

a
2
=
∞
a
=b
( b )]
π  1 [ (b)
a>b
b K a −E a
−1
I01
(a, b, 0) =
10
EDUARDO KAUSEL
∫
AND
MIRZA M. IRFAN BAIG
∞
J1 (ax) J1 (bx)
dx
x2
0
)
(
)
]
1 [( 2
a + b2 (a + b) E (κ) − a2 − b2 (a − b) K (κ)
=
3πab
) ( ) (
) ( )]
 1 [( 2
a + b2 E ab − b2 − a2 K ab
a<b
2  a
=
2b
) (b) ( 2
) ( b )] a = b
3π  1 [( 2
2
2
a
+
b
E
−
a
−
b
K a
a>b
b
a
−2
(a, b, 0) =
I11
Table 3: Integrals for a > 0, b > 0, s > 0, in ENS format. Only the integrals
provided by ENS are included in this table. For a more complete list, see the next table.
√
2 ab
4ab
κ= √
ν=
2
2
(a + b)
(a + b) + s2
K = K (κ) ,
E = E (κ) ,
Λ = Λ (ν, κ) =
|a − b|
s
√
Π (ν, κ)
a+b
2
(a + b) + s2
sκ3
√ E
4π (1 − κ2 ) ab ab
[ (
]
)
2
2
2
2
κ
a
−
b
−
s
κ
1
√
I10
=
E+K
4 (1 − κ2 ) ab
2πa ab
[(
)
]
1 − 12 κ2
sκ
1
√
E−K
I11 =
1 − κ2
2π ab ab
κ
0
= √ K
I00
π ab
1 κs
sgn (a − b)
1
0
√ K−
=−
I10
Λ + H (a − b)
πa 2 ab
πa
a
[(
)
]
2
0
√
=
1 − 12 κ2 K − E
I11
π κ ab
[ √
]
( 2
) κ
1 2 ab
s
s
−1
2
√ K +
I10 =
E+ a −b
sgn (a − b) Λ − H (a − b)
πa
κ
πa
a
2 ab
[
]
)
s
1
κ ( 2
−1
I11
= √
E−
a + b2 + 12 s2 K
2ab
π ab κ
)
[
(
]}
1 { 2
+
a + b2 + a2 − b2 sgn (a − b) π2 Λ − 1
4ab
1
I00
=
(ENS–4.3)
(ENS-4.8)
(ENS-4.4)
(ENS-4.1)
(ENS-4.7)
(ENS-4.2)
(ENS-4.6)
(ENS-4.9)
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
11
Table 4: Integrals for a > 0, b > 0, s > 0, in HP format.
√
√
2
2
A = (a + b) + s2 , B = (a − b) + s2 , L1 = 12 (A − B) ,
L2 = 21 (A + B)
b
a
nba = k a ,
X = checked against
k = L1 /L2 ,
nab = k b ,
available ENS forms
K = K (k) ,
E = E (k) ,
Πab = Π (nab , k) , Πba = Π (nba , k)
[
]
2s
2
1
I00
=
E−K
π L32 (1 − k 2 ) 1 − k 2
[
]
( 2
)
2
a2 − b2 − s2
1
2
I10 =
L2 − a K +
E
πaL32 (1 − k 2 )
1 − k2
[
]
2s
1 + k2
1
E
−
K
I11
=
πabL2 (1 − k 2 ) 1 − k 2
[
]
2s
2
4s
1
I20
=
K
−
E
+
(Πab − K) ,
a<b
3
2
2
πL2 (1 − k )
1−k
π a2 L2
[
]
(
)
2s
2
πL2
4s
=
K
−
E
−
Π
+
ba , a > b
πL32 (1 − k 2 )
1 − k2
π a2 L2
2s
[ (
{
] }
)
)
)
( 2(
(
)
a2 s2 + a2 − b2
2
2
2
2
1
2
L2 2 − k − a K +
E
I21 =
− 2L2 1 − k
πa2 b L2 (1 − k 2 )
L22 (1 − k 2 )
[
]}
{
2s L2
2
k2
1
I22 =
E−K
2 (E − K) +
πa2 b2
1 − k2 1 − k2
2
0
I00
=
K
π L2
2s
0
I10
=
(Πab − K) ,
a<b
πa L2
(
)
2s
π L2
=
− Πba , a > b
πa L2
2s
2
0
I11
=
(K − E)
π L1
]
[
2s2
2L2
a2
0
K
+
(K
−
Π
)
,
I20
=
2
(E
−
K)
+
a<b
ab
πa2
L22
L22
(
[
)]
2L2
a2
2s2
πL2
=
2
(E
−
K)
+
K
+
Π
−
, a>b
ba
πa2
L22
L22
2s
[
]
b2
2s
0
E − K + 2 (Πab − K) ,
a<b
I21 =
πaL1
L2
[
(
)]
2s
b2 πL2
=
E−K+ 2
− Πba , a > b
πaL1
L2
2s
{
}
2
2
0
I22 =
(K − E) + k (K − 2E)
3πL1 k
X
X
X
X
X
X
12
EDUARDO KAUSEL
MIRZA M. IRFAN BAIG
]
a2
s2
E − K + 2 K + 2 (K − Πab ) ,
a<b
L2
L2
X
[
)]
(
2L2
a2
s2
πL2
=
E − K + 2 K + 2 Πba −
, a>b
πa
L2
L2
2s
[
)]
(
s
a2 π L2
b2
−1
I11 =
− Πab , a < b
E − K + 2 (Πab − K) + 2
πL1
L2
L2
2s
X
)]
[
(
s
a2
b2 πL2
− Πba , a > b
=
E − K + 2 (Πba − K) + 2
πL1
L2
L2
2s
[
]
s L2
a2
a2 − b2 + 2s2
−1
I20
=
3
(K
−
E)
−
K
+
(Π
−
K)
a<b
ab
πa2
L22
L22
[
)]
(
s L2
a2
a2 − b2 + 2s2 πL2
=
3
(K
−
E)
−
−
Π
, a>b
K
+
ba
πa2
L22
L22
2s
[
]
( 2
)
2
a2 b2
3b2 s2
−1
I21
=
2b − a2 − s2 (E − K) + 2 K +
(K
−
Π
)
,
a<b
ab
3πa L1
L2
L22
(
[
)]
( 2
)
3b2 s2
πL2
2
a2 b2
2
2
Πba −
=
2b − a − s (E − K) + 2 K +
, a>b
3πaL1
L2
L22
2s
[
s
3b4
a2 b2
−1
I22
=
(5a2 + 5b2 + 2s2 )(E − K) + 2 K + 2 (Πab − K)
6πabL1
L2
L2
(
)]
4
3a
πL2
+ 2
− Πab ,
a<b
L2
2s
[
3a4
s
a2 b2
=
(5a2 + 5b2 + 2s2 )(E − K) + 2 K + 2 (Πba − K)
6πabL1
L2
L2
(
)]
3b4 πL2
+ 2
− Πba ,
a>b
L2
2s
[
3b2 s2
4a2 b2
1
−2
K+
(K − Πab )
I11
(2a2 + 2b2 − s2 )(E − K) +
=
2
3πL1
L2
L22
(
)]
3a2 s2
πL2
+ 2
Πab −
a<b
,
L2
2s
[
3a2 s2
1
4a2 b2
K+
(K − Πba )
=
(2a2 + 2b2 − s2 )(E − K) +
2
3πL1
L2
L22
)]
(
3b2 s2
πL2
+ 2
Πba −
,
a>b
L2
2s
−1
I10
2L2
=
πa
[
AND
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
−2
I20
−2
I21
−2
I22
13
[
L2
a2 (6a2 − 2b2 + 3s2 )
2
2
2
=
(8a
−
4b
+
11s
)(E
−
K)
+
K
9πa2
L22
]
3s2 (3a2 − 3b2 + 2s2 )
+
(K − Πab ) ,
a<b
L22
[
L2
a2 (6a2 − 2b2 + 3s2 )
2
2
2
=
(8a
−
4b
+
11s
)(E
−
K)
+
K
9πa2
L22
)]
(
πL2
3s2 (3a2 − 3b2 + 2s2 )
,
Π
−
a>b
+
ba
L22
2s
[
s
a2 (3a2 + 5b2 )
=
(5a2 − 13b2 + 2s2 )(E − K) −
K
12πaL1
L22
]
3πL2 a4
3(a2 − b2 )2 − 12b2 s2
+
a<b
+
(K
−
Π
)
,
ab
L22
2sL22
[
s
a2 (3a2 + 5b2 )
=
(5a2 − 13b2 + 2s2 )(E − K) −
K
12πaL1
L22
(
]
)
πL2
3πL2 a4
3(a2 − b2 )2 − 12b2 s2
Πba −
,
+
a>b
+
L22
2s
2sL22
{[ (
)2
(
)]
1
=
8 a2 − b2 + 8a2 b2 − s2 9a2 + 9b2 + 2s2 (E − K)
30π ab L1
(
)
(
)}
4 a2 + b2 − s2
πL2
b4 s2
a4 s2
K
+
(K
−
Π
)
Π
−
+ a2 b2
15
+
15
, a<b
ab
ab
L22
L22
L22
2s
{[ (
)2
(
)]
1
=
8 a2 − b2 + 8a2 b2 − s2 9a2 + 9b2 + 2s2 (E − K)
30π ab L1
( 2
)
(
)}
2
− s2
a4 s2
b4 s2
πL2
2 24 a + b
+a b
K
+15
(K
−
Π
)
+
15
Π
−
, a>b
ba
ba
L22
L22
L22
2s
14
EDUARDO KAUSEL
AND
MIRZA M. IRFAN BAIG
Appendix A. Proof of transformation formula. Eason, Noble and Sneddon define the scaled Heuman function
√
(1 − ν) (ν − κ2 )
s
|a − b|
√
√
Λ (ψ, κ) =
Π (ν, κ) =
Π (ν, κ)
(a + b)
ν
2
(A.1)
(a + b) + s2
= [E (κ) − K (κ)] F (ψ, κ′ ) + K (κ) E (ψ, κ′ )
where
1
sin ψ = ′
κ
√
ν − κ2
s
= ,
ν
B
cos ψ =
κ
κ′
√
1−ν
|a − b|
=
,
ν
B
tan ψ =
s
|a − b|
(A.2)
Thus, we need expressions to move from one set of parameters to the other. From
Gradshteyn and Ryzhik, page 907, formulas 8.121–3, 8.121–4, and page 908, formulas
8.125 & 8.126, the Landen-Gauss transformations which are relevant to this proof are:
√
2 k
2
κ=
,
E (k) − (1 − k) K (k)
K (κ) = (1 + k) K (k) ,
E (κ) =
(A.3a)
1+k
1+k
κ′ =
E (ψ, κ′ ) =
1−k
,
1+k
F (ψ, κ′ ) = (1 + k) F (φ, k ′ )
2
1−k
[E (φ, k ′ ) + k F (φ, k ′ )] −
sin ψ
1+k
1+k
(A.3b)
where
tan (ψ − φ) = k tan φ
(A.3c)
Substituting these into the expansion of the scaled Heuman function above, we obtain
Λ (ψ, κ) = [E (κ) − K (κ)] F (ψ, κ′ ) + K (κ) E (ψ, κ′ )
[
]
2
=
E (k) − (1 − k) K (k) − (1 + k) K (k) (1 + k) F (φ, k ′ )
1+k
{
}
2
1−k
′
′
+ (1 + k) K (k)
[E (φ, k ) + k F (φ, k )] −
sin ψ
1+k
1+k
= 2 {[E (k) − K (k)] F (φ, k ′ ) + K (k) E (φ, k ′ )} − (1 − k) sin ψ K (k)
But
(1 − k) sin ψ =
so
(
)
L1
s
s
1−
=
L2 L2 − L1
L2
Λ (ψ, κ) = 2Λ (φ, k) −
s
K (k)
L2
(A.4)
(A.5)
(A.6)
Also, expanding the expression for tan (ψ − φ), we obtain
k tan ψtan2 φ + (1 + k) tan φ − tan ψ = 0
which is a quadratic equation in tan φ. Its solution is
{
}
√
1+k
4k
2
tan φ = −
1∓ 1+
2 tan ψ
2k tan ψ
(1 + k)
{
}
√
1
2 tan2 ψ
1
∓
1
+
κ
=−
(1 − κ′ ) tan ψ
(A.7)
(A.8)
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
But
κ tan ψ =
and
√
v
u
u
1 + κ2 tan2 ψ = t1 + [
√
1 − κ′ 2
s
|a − b|
15
(A.9)
4abs2
]
2
2
(a + b) + s2 (a − b)
v
u
2
2
2
u (a + b) (a − b) + s2 (a − b) + 4abs2
[
]
=t
2
2
(a + b) + s2 (a − b)
√
2
(a − b) + s2
a+b
a+b ′
√
=
=
κ
|a − b|
|a − b|
2
(a + b) + s2
Hence, taking into account that k = (1 − κ′ ) / (1 + κ′ ), then
{
}
a+b ′
|a − b|
1
1
±
tan φ = −
κ
=
{− |a − b| ∓ (a + b) κ′ }
′
(1 − κ ) s
|a − b|
(1 − κ′ ) s
{
sgn (a − b)
bk − a
=
b − ak
ks
(A.10)
(A.11)
A) Let’s consider the first of the two solutions above. In this case
tan φ =
s2 kb
s
bk − a
sgn (a − b) = 2
sgn (a − b) =
sgn (a − b) (A.12)
2
ks
(L1 − b ) ks
b (nab − 1)
so
tan ψ
=
tan φ

 >1 a<b
1 − k ab
1 − nab
=
=
→
±∞ a = b
s

1 − ab
1 − ab
b(nab −1) sgn (a − b)
<0 a>b
s
|a−b|
(A.13)
Case 1: a < b
Here tan ψ > tan φ i.e. ψ > φ and ψ − φ > 0. Also, k tan φ > 0, in which case the
angle transformation
formula tan (ψ − φ) = k tan φ is satisfied with positive angles in
[ 1 ]
the range 0, 2 π . Thus, in this case
√
1 nab − k 2
s
tan φ =
=
(A.14a)
k
1 − nab
b (1 − nab )
√
1 nab − k 2
s
sin φ = ′
= ′ √
(A.14b)
k
nab
k L2 1 − nab
√
√
k 1 − nab
b 1 − nab
cos φ = ′
=
(A.14c)
k
nab
k ′ L2
Also
√
(1 − nab ) (nab − k 2 )
s
Λ (φ, k) ≡ Λab =
Π (nab , k) =
Πab
(A.15)
nab
L2
so
s
Λ (ψ, κ) =
[2Πab − K] ,
a<b
(A.16)
L2
16
EDUARDO KAUSEL
AND
MIRZA M. IRFAN BAIG
Case 2: a > b
Now tan φ < 0, i.e. φ < 0, which we reject.
B) Consider next the second solution for tan φ:
tan φ =
b − ak
sa
s
sgn (a − b)
sgn (a − b) = 2
sgn (a − b) =
2
ks
a − L1
a (1 − nba )
so
s
|a−b|
tan ψ
=
tan φ
s
a(1−nba )
=
sgn (a − b)
1 − k ab
1 − nba
=
1 − ab
1−
b
a

 <0 a<b
→
±∞ a = b

>1 a>b
(A.17)
(A.18)
We see that when a < b we obtain a negative solution φ < 0, so we reject it as well. On
the other hand, for a > b this leads us to
√
1 nba − k 2
s
tan φ =
=
(A.19a)
k
1 − nba
a (1 − nba )
√
1 nba − k 2
s
sin φ = ′
(A.19b)
= ′ √
k
nba
k L2 1 − nba
√
√
k 1 − nba
a 1 − nba
cos φ = ′
=
(A.19c)
k
nba
k ′ L2
and
√
(1 − nba ) (nba − k 2 )
s
Λ (φ, k) ≡ Λba =
Π (nba , k) =
Πba
(A.20)
nba
L2
so
s
Λ (ψ, κ) =
[2Πba − K] ,
a>b
(A.21)
L2
An additional useful formula is derived next. From section 3, the characteristics of
Πab , Πba are nab = k ab > k 2 and nba = k ab > k 2 , which together satisfy the relationship nab nba = k 2 . Hence, the special addition formula 117.02 on page 13 in Byrd and
Friedman applies to these functions:
√
( a )
( b )
k ab
π
(
)
)
(
Π k b , k + Π k a , k = K (k) +
2
1 − k ab k ab − k 2
(A.22)
√
1
π
(
)
= K (k) +
2 k 2 − ab + ab k + 1
Also
k
so
(a
b
+
b
a
)
=
L1 a2 + b2
a2 + b2
L2 + L22 − s2
=
= 1
2
L2 ab
L2
L22
√
(
)
1 − k ab + ab + k 2 =
√
and k 2 =
L2
L2 + L22 − s2
+ 12 =
1− 1
2
L2
L2
√
L21
L22
s2
s
=
L22
L2
(A.23)
(A.24)
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
17
Hence
Πab + Πba = K (k) +
πL2
2s
(A.25)
Appendix B. Derivatives and continuity of Lambda function. It can be shown
that
(
)
∂Π (nab , k)
∂Πab ∂nab
∂Πab ∂k
∂Πab k a ∂k
∂Πab ∂k
=
+
=
+
+
∂a
∂nba ∂a
∂k ∂a
∂nab b
b ∂a
∂k ∂a
)
( )2
(
)(
1 − ab
b
∂k
a
=
k+a
K (k) +
E (k) − bk
Πab
2ka (ka − b)
∂a
kb − a
kb − a
(
)
b
E (k) ∂k
+
Πab −
a − kb
1 − k 2 ∂a
(B.1a)
(
)
∂Πba
∂Πba ∂nba
∂Πba ∂k
b
∂k
∂Πba
∂Πba ∂k
=
+
= 2 a
−k
+
∂a
∂nba ∂a
∂k ∂a
a
∂a
∂nba
∂k ∂a
)
( )2
(
)(
1 − ab
1
∂k
b
(B.1b)
=
a
−k
K (k) +
E (k) − ak
Πba
2k (kb − a)
∂a
ak − b
ak − b
(
)
E (k) ∂k
a
Πba −
+
b − ak
1 − k 2 ∂a
When a = b, then
)(
(
)
∂Πab ∂k 1
1
k
+
b
E
(k)
−
K
(k)
=
∂a a=b
2kb (1 − k)
∂a a=b
1−k
(B.2a)
)
(
1 ∂k E (k)
+
Π
−
ab
1 − k ∂a a=b
1 − k2
(
)
)(
∂k ∂Πba 1
1
b
E
(k)
−
K
(k)
=
−
k
∂a a=b
2kb (1 − k)
∂a a=b
1−k
(B.2b)
)
(
1 ∂k E (k)
+
Πba −
1 − k ∂a a=b
1 − k2
and from their difference
(
)
(
)
∂Πab
∂Πba 1
1
1 ∂k −
=
E
(k)
−
K
(k)
+
(Πab − Πba )|a=b
∂a
∂a a=b
b (1 − k) 1 − k
1 − k ∂a a=b
(B.3)
Since (Πab − Πba )|a=b = 0, then
(
)
(
)
∂Πab
∂Πba 1
1
̸= 0
−
E
(k)
−
K
(k)
(B.4)
=
∂a
∂a
b
(1
−
k)
1
−
k
a=b
a=b
which shows that Πab , Πba do not continue one into the other at a = b, but have distinct
slopes at this transitional value.
Consider next the ENS Lambda function:
√
(1 − ν) (ν − κ2 )
√
Π (ν, κ) = π2 Λ0 (ψ\ϑ)
(B.5)
Λ (ν, κ) =
ν
18
EDUARDO KAUSEL
AND
MIRZA M. IRFAN BAIG
When a = b, then ν = 1 with κ < 1 if s √
> 0, in which case Π (1, κ) = ∞. However,
Λ (ν, κ) remains finite because of the factor (1 − ν) → 0. The proof of this relies on the
relationship between the ENS Lambda function and Heuman’s function. Indeed, when
ν = 1 → cos2 ψ = 0 → ψ = 12 π, we have
)
)
(
(
for any κ < 1
Λ (1, κ) = 12 π Λ0 12 π\ϑ = π2 × 1 = π2 ,
ϑ < 12 π ,
a=b
so the Lambda function remains finite at a = b. On the other hand, in Appendix A we
found the equivalence
{
{
s
Λab
2Πab − K (k) , a < b
Λ (ν, κ) =
=
(B.6)
Λba
2Πba − K (k) , a > b
L2
which we have verified to be true by numerical testing. Since Πab |a=b = Πba |a=b ,
it follows that Λab |a=b = Λba |a=b = Λ (1, κ) = 12 π, so the Lambda function itself is
continuous at a = b. However, as we have already found out in B.4
∂Πab ∂Πba ̸=
(B.7)
∂a ∂a a=b
a=b
Hence, Λ exhibits a discontinuity of slope when a = b, so it is discontinuous. On the
other hand, the modified function
a−b
s
sgn (a − b) Λ (ν, κ) =
Π (ν, κ)
(B.8)
a + b L2 (1 + k)
jumps from −1 through 0 to +1 in the neighborhood a = [b − ε, b, b + ε]. Hence, the
equivalence
{
s
K (k) − 2Πab , a < b
(B.9)
sgn (a − b) Λ (ν, κ) =
2Πba − K (k) , a > b
L2
is intrinsically discontinuous, so HP’s use of the upper expression in the domain a > b lead
them necessarily to an erroneous branch. Nonetheless, despite the intrinsic discontinuity
of the Lambda function, many of the integrals and particularly those that affect HP’s
elasticity problem are still continuous up to first order, i.e. strains and stresses are
continuous. For example, in eq. 5.1b, the discontinuous fragment is
[
]
f (a, b, s) = sgn (a − b) π2 Λ (ν, κ) − 1
(B.10)
Since as we have just seen π2 Λ (1, κ) = 1, then f (a, a, s) = sgn (a − b) × 0 = 0, both when
approaching from the left or from the right, so the primitive fragment is continuous.
Also, the first derivative will contain a term of the form
[2
]
(B.11)
π Λ (ν, κ) − 1 δ (a − b)
which despite the Dirac-delta factor is also zero at the transition, so again the first derivative of the fragment is continuous. Higher derivatives, however, will be discontinuous.
The previous results lead to an interesting corollary. Since at a = b Λaa = 21 π and
Πaa = Π (k, k), then
L2
2Π (k, k) − K (k) = π
(B.12)
2s
where
(√
)
(√
)
( s )2
( s )2
s
s
L1 = b
1+
−
, L2 = b
1+
+
(B.13)
2b
2b
2b
2b
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
and
(
)
2s
s b
L2 − L1
L1
=2
=2
=2 1−
= 2 (1 − k)
L2
b L2
L2
L2
k=
L1
L2
b2
= 21 = 2
L2
b
L2
so
2Π (k, k) − K (k) =
19
(B.14)
(B.15)
π
2 (1 − k)
(B.16)
π
2
(B.17)
or
(1 − k) [2Π (k, k) − K (k)] =
which is valid for any 0 < k < 1. Of course, this is nothing but an alternative representation of Heuman’s lambda function satisfying the limiting condition
(
)
Λ0 (ν, κ)|ν=1 = Λ0 12 π\ϑ = 1
(B.18)
Appendix C. Numerical integration. To avoid mistakes, all of the integration
formulas given earlier have been verified by direct numerical integration. To this purpose,
the improper integrals are divided into two integration ranges [0, x0 ] and [x0 , ∞], where
x0 defines the start of the tail. The body is integrated by an appropriate numerical
quadrature which accounts for the rate of change of the integrands, while the tail is
obtained in closed-form using asymptotic expansions as follows.
√
]
[
2
Jm (ax) ≈
(C.1)
cos ax − 41 π (1 + 2m)
πax
√
[
]
2
Jn (bx) ≈
cos bx − 14 π (1 + 2n)
(C.2)
πbx
[
]
[
]
2 1
√ cos ax − 41 π (1 + 2m) cos bx − 41 π (1 + 2n)
Jm (ax) Jn (bx) ≈
(C.3)
π x ab
But
cos φ cos ψ = 21 {cos (φ − ψ) + cos (φ + ψ)}
(C.4)
so
]
[
]
[
cos ax − 14 π (1 + 2m) cos bx − 41 π (1 + 2n)
{
[
]
[
]}
= 21 cos (a − b) x − 21 π (m − n) + cos (a + b) x − 12 π (m + n + 1)
{
= 12 cos (a − b) x cos 12 π (m − n) + sin (a − b) x sin 12 π (m − n)
(C.5)
}
+ cos (a + b) x cos 12 π (m + n + 1) + sin (a + b) x sin 12 π (m + n + 1)
Also
{
cos
1
2 π (m
− n) =
{
cos
1
2 π (m
+ n + 1) =
1
(m−n)
(−1) 2
0
1
(−1) 2
m − n is even
m − n is odd
0
m + n is even
(m+n+1)
m + n is odd
(C.6a)
20
EDUARDO KAUSEL
AND
MIRZA M. IRFAN BAIG
{
sin 12 π (m − n) =
{
sin
1
2 π (m
+ n + 1) =
0
m − n is even
1
(m−n−1)
(−1) 2
m − n is odd
1
(m+n)
(−1) 2
0
(C.6b)
m + n is even
m + n is odd
Then again, m ± n is even if both m, n are even, or both are odd, and it is odd if one is
even and the other one is odd. Hence,
[
]
1
1
1
(m−n)
(m+n)
2
2
√
cos (a − b) x + (−1)
sin (a + b) x
Jm (ax) Jn (bx) ≈
(−1)
(C.7a)
πx ab
even m ± n
Jm (ax) Jn (bx) ≈
1
√
πx ab
[
]
1
1
(m−n−1)
(m+n+1)
(−1) 2
sin (a − b) x + (−1) 2
cos (a + b) x
odd m ± n
(C.7b)
1
(m+n)
1
(m−n)
n
1
(m−n)
Observe that (−1) 2
= (−1) 2
(−1) ̸= (−1) 2
.
We now define the following exponential integrals:
∫ ∞
s cos Ax0 − A sin Ax0 −sx0
C0 (A, s) =
e−sx cos A x dx =
e
(C.8a)
A2 + s2
x0
∫ ∞
A cos Ax0 + s sin Ax0 −sx0
e
(C.8b)
S0 (A, s) =
e−sx cos A x dx =
A2 + s2
x0
∫ ∞
cos A x
C−1 (A, s) =
e−sx
dx = Re {E1 [x0 (s + i A)]}
(C.8c)
x
x0
∫ ∞
sin A x
dx = − Im {E1 [x0 (s + i A)]}
(C.8d)
S−1 (A, s) =
e−sx
x
x0
∫ ∞
cos A x
e−sx0 cos A x0 − Re {x0 (s + iA) E1 [x0 (s + iA)]}
C−2 (A, s) =
e−sx
dx
=
x2
x0
x0
(C.8e)
∫ ∞
−sx0
sin A x
e
sin Ax0 + Im {x0 (s + iA) E1 [x0 (s + iA)]}
S−2 (A, s) =
e−sx
dx =
2
x
x0
x0
(C.8f)
where E1 (z)= exponential integral, which can readily be evaluated using Matlab. We
λ
can now express the tail Tmn
(a, b, s) of the integrals as:
Even m ± n:
∫
∞
e−sx xλ Jm (ax) Jn (bx) dx
[
]
1
1
1
(m−n)
(m+n)
= √
(−1) 2
Cλ−1 (a − b, s) + (−1) 2
Sλ−1 (a + b, s)
π ab
(C.9a)
λ
Tmn
(a, b, s) =
x0
LAPLACE TRANSFORM OF PRODUCTS OF BESSEL FUNCTIONS
Odd m ± n:
∫
21
∞
e−sx xλ Jm (ax) Jn (bx) dx
x0
[
]
1
1
1
(m−n−1)
(m+n+1)
2
2
(−1)
Sλ−1 (a − b, s) + (−1)
Cλ−1 (a + b, s)
= √
π ab
(C.9b)
λ
Tmn
(a, b, s) =
References
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Graphs, and Mathematical Tables, U.S. National Bureau of Standards, 1972.
[2] P.F. Byrd and M.D. Friedman. Handbook of Elliptic Integrals for Engineers and Scientists, 2nd
Edition (revised), Springer–Verlag, 1971.
[3] G. Eason, B. Noble and I.N. Sneddon. On certain integrals of Lipschitz-Hankel type involving
products of Bessel functions, Philosophical. Transactions of the Royal Society of London, Series A,
247, p. 529–551, 1955.
[4] I.S. Gradshteyn and Ryzhik. Table of Integrals, Series and Products, Academic Press, 1980.
[5] M.T. Hanson and I.W. Puja. The evaluation of certain infinite integrals involving products of Bessel
functions: a correlation of formula, Quarterly of Applied Mathematics, 55 (3), p. 505–524, 1997.
[6] A.E.H. Love. (1929). The stress produced in a semi-infinite solid by pressure on part of the boundary,
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[7] F. Oberhettinger. Tables of Laplace transforms, Springer–Verlag, 1973.