integrals Bessel

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An integral containing a Bessel Function and a Modified
Bessel Function of the First Kind
Amelia Carolina Sparavigna
To cite this version:
Amelia Carolina Sparavigna. An integral containing a Bessel Function and a Modified Bessel Function
of the First Kind. 2017. <hal-01564567>
An integral containing a Bessel Function and a Modified Bessel Function of the 
First Kind 
 
Amelia Carolina Sparavigna 
Politecnico di Torino, Italy 
 
Here we discuss the calculation of an integral containing the Bessel function J0(r) and the modified 
Bessel function I1(r) of the first kind. The method of the calculus is based on some products of the 
Bessel and modified Bessel functions and their derivatives. The method could be useful for training 
the students in the manipulation of such integrals. 
 
Here we are proposing a method for the calculus of an integral of the form ∫ drrIrJr )()( 10
2 . The 
method is based on some suitable functions, which are the products of the Bessel and modified 
Bessel functions and their derivatives. The approach, which is coming from the author’s experience, 
could be useful for training the students in the manipulation of integrals containing Bessel 
functions. The author’s experience was concerning a calculus required by a method for determining 
the thermal diffusivity, based on the measurement of the thermal expansion of a cylindrical sample 
[1-4]. For this method, I had to consider the problem of the thermoelasticity. In particular, I had to 
test if the thermal expansion of a cylindrical sample, assumed as the integral of temperature 
),,( tzrθ , that is ∫=∆
l
dztzr
0
),,(θβ , ( β is the coefficient of the expansion), was equal to the 
component zu of the displacement field u
r
 found by means of the thermoelasticity equations. These 
equations are [5]: 
 
ugradudivgradu
rrr
ρθβµλµλµ =+−++∇ )23()()(2 
0)(
)(
2 =
−
−−∇ udiv
cc
ck
vp
v
r&
β
θθ
 
 
λ and µ are the Lamè coefficients. pc and vc are the specific heats at constant pressure and 
volume, k is the thermal conductivity and ρ the density. In the case of a solid material, the second 
equation reduces to 02 =−∇ θθ &vck . The solution of these equations for a cylindrical sample is 
requiring hyperbolic sines and cosines and Bessel functions )(0 rJ and )(1 rI , and the related 
integrals. Here we consider and discuss how to calculate one of these integrals, that of the form: 
∫ drrJrIr )()( 01
2 . 
Let us start from the equation of the Bessel function and of the modified Bessel function of the first 
kind (the variable r is here assumed as dimensionless): 
 
0)()(')('' 0
2
00
2 =++ rJrrJrrJr (1) 
0)()1()(')('' 1
2
11
2 =+−+ rIrrIrrIr
 
(2) 
 
We can multiply (1) by )(1 rI and (2) by )(0 rJ and then subtract the results as follow: 
 
0)()()1(
)()(')()('')()()()(')()(''
10
2
0101
2
10
2
1010
2
=++
−−++
rIrJr
rJrIrrJrIrrIrJrrIrJrrIrJr
 
 
0)()()12()]()(')()('[)]()('')()(''[ 10
2
01100110
2 =++−+− rIrJrrJrIrIrJrrJrIrIrJr
 
(3)
 
 
Let us consider the function W and its derivative, defined in the following manner: 
 
)('')()()(''
)('')()(')(')(')(')()('''
)(')()()('
1010
10101010
1010
rIrJrIrJ
rIrJrIrJrIrJrIrJW
rIrJrIrJW
−=
−−+=
−=
 
 
Therefore, from (3), we have: 
)()()12(2'
0)()()12('
10
22
10
22
rIrJrrWrWWr
rIrJrrWWr
+−=+
=+++
 
 
A simple integration gives:
 
 
drrIrJdrrIrJrdrrWWr ∫∫∫ −−= )()()()(2 1010
22
 
(4) 
 
Moreover, we have the derivatives of the Bessel functions [6]: 
 
r
rI
rIrIrIrI
r
rJ
rJrJrJrJ
)(
)()(';)()('
)(
)()(';)()('
1
0110
1
0110
−==
−=−=
 (5)
 
From (4), using (5): 
 
drrIrJdrrIrJrdrrrIrJrIrJrIrJr
drrIrJdrrIrJrdrrWWr
∫∫∫
∫∫∫
++−+=
++−=−
)()()()(2]/)()()()()()([
)()()()(2
1010
2
100011
1010
22
 
 
∫∫ ++=− drrIrJrdrrIrJrIrJrWr )()(2)]()()()([ 10
2
0011
2
 
Therefore; 
 
∫∫ +−−= drrIrJrIrJrWrdrrIrJr )]()()()([
2
1
2
1
)()( 0011
2
10
2
 (6)
 
 
Then, to evaluate the integral in the left side of (6), we need the integrals
 
∫ drrIrJr )()( 11 and 
∫ drrIrrJ )()( 00 . These integrals are easy to calculate. 
Let us consider the two functions )()(;)()( 012101 rIrJrrIrJr =Π=Π . Let us evaluate the 
following: 
 
)()()()(
)()(/)()()()()()('
)()()()(
/)()()()()()()()('
1100
110100012
0011
100011101
rIrJrrIrJr
rIrJrrrIrJrrIrJrrIrJ
rIrJrrIrJr
rrIrJrrIrJrrIrJrrIrJ
+=
+−+=Π
+−=
−+−=Π
 
 
Adding ',' 21 ΠΠ , we obtain: 
 
)()(2
)()()()()()()()(''
00
1100001121
rIrJr
rIrJrrIrJrrIrJrrIrJr
=
+++−=Π+Π
 
 
As a consequence (after an integration): 
 
→Π+Π=∫ 2100 )()(2 drrIrrJ )]()()()([
2
)()( 011000 rIrJrIrJ
r
drrIrJr +=∫
 (7) 
 
 
Subtracting ',' 21 ΠΠ , we obtain: 
 
 
)()(2
)()()()()()()()(''
11
1100001121
rIrJr
rIrJrrIrJrrIrJrrIrJr
−=
−−+−=Π−Π
 
Therefore, we have: 
→Π+Π−=∫ 2111 )()(2 drrIrrJ )]()()()([
2
)()( 011011 rIrJrIrJ
r
drrIrJr +−=∫
 (8) 
Using the results (7) and (8): 
 
)]()()()([
2
)]()()()([
2
1
)()(
2
]/)()()()()()([
2
1
)]()()()([
4
)]()()()([
4
)](')()()('[
2
1
)()(
01101100
2
01100011
2
01100110
1010
2
10
2
rIrJrIrJ
r
rIrJrIrJr
rIrJ
r
rrIrJrIrJrIrJr
rIrJrIrJ
r
rIrJrIrJ
r
rIrJrIrJrdrrIrJr
+−+=
−+−−−=
+−−+
−−=∫
 
 
As we have seen in the calculation, to find the integral ∫ drrIrJr )()( 10
2 we used functions 
21,, ΠΠW
 
and their derivatives. The search for these functions could be intriguing to students, and 
for this reason, it could stimulate their interest during the study of the integrals of Bessel functions. 
 
References 
[1] Omini, M., Sparavigna, A., & Strigazzi, A. (1990). Dilatometric determination of thermal 
diffusivity in low conducting materials. Measurement Science and Technology, 1(2), 166. 
[2] Sparavigna, A., Giachello, G., Omini, M., & Strigazzi, A. (1990). High-sensitivity capacitance 
method for measuring thermal diffusivity and thermal expansion: results on aluminum and copper. 
International Journal of Thermophysics, 11(6), 1111-1126. 
[3] Omini, M., Sparavigna, A., & Strigazzi, A. (1990). Calibration of capacitive cells for 
dilatometric measurements of thermal diffusivity. Measurement Science and Technology, 1(11), 
1228. 
[4] Sparavigna, A. C. (1990). Nuovo metodo dilatometrico di misura della diffusività termica dei 
solidi (Tesi di Dottorato). 
[5] Kovalenko, A. D. (1970). Thermoelasticity. Basic theory and applications. Wolters-Noordhoff. 
[6] Abramowitz, M., & Stegun, I. A. (1972). Handbook of mathematical functions, National Bureau 
of Standards, Applied Mathematics Series – 55, 1972, Chapter 9, Bessel function of Integer Order, 
F. W. J. Oliver.