An integral containing a Bessel Function and a Modified Bessel Function of the First Kind

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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:

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

∫

r2J₀(r)I₁(r)dr. 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

) , , (r z t θ , that is ∆=

∫

l r z t dz 0

θ

( , , )

β

, (β is the coefficient of the expansion), was equal to the component u_zof the displacement field ur found by means of the thermoelasticity equations. These equations are [5]: u grad u div grad ur

λ

µ

r

λ

µ

β

θ

ρ

r

µ

∇2 +( + ) ( )−(3 +2 ) = 0 ) ( ) ( 2 _{− −} − ₌ ∇ c c c div u k _v & p v r

β

θ

θ

λ

and µ are the Lamè coefficients. c and p c are the specific heats at constant pressure and v

integrals. Here we consider and discuss how to calculate one of these integrals, that of the form:

∫

r2I₁(r)J₀(r)dr.

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 ₀ 0 2 _{+ + =} r J r r J r r J r (1) 0 ) ( ) 1 ( ) ( ' ) ( ' ' ₁ 2 ₁ 1 2 _{+ − + =} r I r r I r r I r (2)

We can multiply (1) by I₁(r) and (2) by J₀(r) and then subtract the results as follow:

0 ) ( ) ( ) 1 ( ) ( ) ( ' ) ( ) ( ' ' ) ( ) ( ) ( ) ( ' ) ( ) ( ' ' 1 0 2 0 1 0 1 2 1 0 2 1 0 1 0 2 = + + − − + + r I r J r r J r I r r J r I r r I r J r r I r J r r I r J r 0 ) ( ) ( ) 1 2 ( )] ( ) ( ' ) ( ) ( ' [ )] ( ) ( ' ' ) ( ) ( ' ' [ _{0 1 1 0 0 1 1 0} 2 _{0 1} 2 _{− + − + + =} r I r J r r J r I r I r J r r J r I r I r J r (3)

Let us consider the function W and its derivative, defined in the following manner:

) ( ' ' ) ( ) ( ) ( ' ' ) ( ' ' ) ( ) ( ' ) ( ' ) ( ' ) ( ' ) ( ) ( ' ' ' ) ( ' ) ( ) ( ) ( ' 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 r I r J r I r J r I r J r I r J r I r J r I r J W r I r J r I r J W − = − − + = − =

Therefore, from (3), we have:

) ( ) ( ) 1 2 ( 2 ' 0 ) ( ) ( ) 1 2 ( ' 1 0 2 2 1 0 2 2 r I r J r rW rW W r r I r J r rW W r + − = + = + + +

A simple integration gives:

Moreover, we have the derivatives of the Bessel functions [6]: r r I r I r I r I r I r r J r J r J r J r J ) ( ) ( ) ( ' ; ) ( ) ( ' ) ( ) ( ) ( ' ; ) ( ) ( ' 1 0 1 1 0 1 0 1 1 0 − = = − = − = (5) From (4), using (5): dr r I r J dr r I r J r dr r r I r J r I r J r I r J r dr r I r J dr r I r J r dr rW W r

∫

∫

∫

∫

∫

∫

+ + − + = + + − = − ) ( ) ( ) ( ) ( 2 ] / ) ( ) ( ) ( ) ( ) ( ) ( [ ) ( ) ( ) ( ) ( 2 1 0 1 0 2 1 0 0 0 1 1 1 0 1 0 2 2

∫

∫

+ + = −r2W r[J₁(r)I₁(r) J₀(r)I₀(r)]dr 2 r2J₀(r)I₁(r)dr Therefore;

∫

∫

r J r I r dr =− r W − r[J (r)I (r)+J (r)I (r)]dr 2 1 2 1 ) ( ) ( ₁ 2 _{1 1 0 0} 0 2 (6)

Then, to evaluate the integral in the left side of (6), we need the integrals

∫

rJ₁(r)I₁(r)dr and

∫

rJ₀(r)I₀(r)dr. These integrals are easy to calculate.

As a consequence (after an integration): → Π + Π =

∫

0( ) 0( ) 1 2 2 rJ r I r dr [ ( ) ( ) ( ) ( )] 2 ) ( ) ( _{0 0 1 1 0} 0 J r I r J r I r r dr r I r J r = +

∫

(7) Subtracting Π₁',Π₂', we obtain: ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ' ' 1 1 1 1 0 0 0 0 1 1 2 1 r I r J r r I r J r r I r J r r I r J r r I r J r − = − − + − = Π − Π Therefore, we have: → Π + Π − =

∫

1( ) 1( ) 1 2 2 rJ r I r dr [ ( ) ( ) ( ) ( )] 2 ) ( ) ( _{1 0 1 1 0} 1 J r I r J r I r r dr r I r J r = − +

∫

(8) Using the results (7) and (8):

)] ( ) ( ) ( ) ( [ 2 )] ( ) ( ) ( ) ( [ 2 1 ) ( ) ( 2 ] / ) ( ) ( ) ( ) ( ) ( ) ( [ 2 1 )] ( ) ( ) ( ) ( [ 4 )] ( ) ( ) ( ) ( [ 4 )] ( ' ) ( ) ( ) ( ' [ 2 1 ) ( ) ( 0 1 1 0 1 1 0 0 2 0 1 1 0 0 0 1 1 2 0 1 1 0 0 1 1 0 1 0 1 0 2 1 0 2 r I r J r I r J r r I r J r I r J r r I r J r r r I r J r I r J r I r J r r I r J r I r J r r I r J r I r J r r I r J r I r J r dr r I r J r + − + = − + − − − = + − − + − − =

∫

As we have seen in the calculation, to find the integral

∫

r2J₀(r)I₁(r)dr we used functions

2 1,

,Π Π

W 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.

[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).