NATIONAL BUREAU OF STANDARDS REPORT

2895

ON A CERTAIN INTEGRAL INVOLVING BESSEL FUNCTIONS

by

H. A. Antosiewicz

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following listing of

the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its title. A brief description

of the activities, and of the resultant reports and publications, appears on the inside of the back

cover of this report.

Electricity. Resistance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Applied Electricity. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Gage.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenics. Engines and

Lubrication. Engine Fuels. Cryogenic Engineering.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State

Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectroscopy.

Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Instruments Branch.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. Capacity, Density, and Fluid Meters. Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Organic Plastics. Dental Research.

Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Concreting Materials. Constitution and Microstructure. Chemistry of Mineral Products.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Machine

Development.

Electronics. Engineering Electronics. Electron Tubes. Electronic Computers. Electronic

Instrumentation.

Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Frequency Utilization Research. Tropospheric Propagation Research. High

Frequency Standards. Microwave Standards.

Ordnance Development. These three divisions are engaged in a broad program of research

Electromechanical Ordnance, and development in advanced ordnance. Activities include

Ordnance Electronics. basic and applied research, engineering, pilot production, field testing, and evaluation of a wide variety of ordnance materiel. Special skills and facilities of other

NBS divisions also contribute to this program. The activity is sponsored by the Department of

Defense.

Missile Development. Missile research and development: engineering, dynamics, intelligence,

instrumentation, evaluation. Combustion in jet engines. These activities are sponsored by the

Department of Defense. • Office of Basic Instrumentation • Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

wis project ms report

1 102-10-1 10L

t- October 26, 1953 2895

ON A CERTAIN INTEGRAL INVOLVING BESSEL FUNCTIONS

by

K. A. Antosiewicz

National Bureau of Standards

The American University

<MS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, Approved for public release by the

unless permission n Director of the National Institute of

25 , d,c, Such pen Standards and Technology (NIST)

caliy prepared If th on October 9, 2015

n part, Is prohibited

indards, Washington

jort has been speclflport for Its own use,

ON A CERTAIN INTEGRAL INVOLVING BESSEL FUNCTIONS*

by

H. A 0 Antosiewicz

This note deals with the integral

oo .2 J (rx)Y (ax)-J (ax)Y (rx) dx kx t o o o o X X

which arises in the solution of the equation of heat flow

in a region internally bounded by a circular cylinder with

constant surface temperature [1, pp. 280-282], Its evaluation for large values of t presents certain difficulties

because the standard Abelian theorems for Laplace integrals

are not applicable. Although asymptotic expansions of

similar integrals can be found in the literature [1,3?*+]?

they generally are stated without explanation of their derivation, It seems desirable, therefore, to bridge this

gap by giving a rather complete discussion, which may

prove useful in the evaluation of related integrals.

The problem was brought to our attention by B. Peavy of

the National Bureau of Standards; we appreciate his interest

in the publication of our result in this form.

1 , Let

*This paper was prepared under a National Bureau of Standards contract with The American University.
-2-

(1 ) F(x) =

and put

(2) f(t)

J (rx)Y (ax)-J (ax)Y (rx)

o o o o

[

J

o 2 (ax)+Y

o 2 (ax) ] ,x

00 = JT r e kx t F(x)dx

where it is assumed that a, k, r are positive constants

From the definitions of the Bessel functions J (z),

o Y (z) we obtain the expressions

( 3 )

(4)

J (rx)Y (ax) - J (ax)Y (rx) = “ ~ [ 1 + 0(x2

)], O O O O ** L

J

2

(ax) + Y

2

(ax) = [1 + (~ £ n ^

9

ax

) 2

][ 1 + 0(x2 )]*

where, as customary, 0(x ) denotes a function such

tha t|pb(x) /x2

< A for some A > 0 as x —^0. Consequently,

we find

( 5 )

F (x) = % in f

[1 + (2/„ -to) 2 ]- 1 ) 1 =

r

1 + O(x^) XL

£ n “ ([1t(/n “^-7p) 2

]x) 1 [ 1 + G(x) ] 2

7T

where the function G(x) is such that G(x) = 0(x ) as x—?0

and G(x) - 0(ex

) as x —?oo # Thus, F(x) has an infinite

discontinuity at x = 0, and the integral in (2) is for

any(finite) value of t an improper integral. However, if

*Throughout , C~=~oT577» « . Euler’s constant
-

3 -

we denote by g(x) the numerator in the second expression of

( 5 ) and note that |g(x)| <_

on any interval 0 <_ x <_X, then

for any sufficiently small > 0

so that the integral in (2) certainly converges at its lower

limit for any value of t _> 0. The convergence at its upper

limit is evident.

It is to be noted that the fact that the integral in (2)

is an improper integral a fortiori precludes the possibility

of using directly numerical methods for its evaluation.

And this, of course, holds true regardless of the value of

t for which (2) is to be calculated.

2. In analogy to the asymptotic theory of functions

defined by Laplace integrals it can be shown that the important contribution to f(t) for large values of t will

come from the values of F(x) near x = 0. Hence we need to

consider the two integrals, obtained by substituting the

expression (5) into (2), namely.
-J+-

00

f(t) = hi*b IT

a (* n — [ r jo

r

kx 1

[1 + (§ in 4

ai ) l \ 2 ]- 1 ^ ir 2

J x

( 7 )

oo + ^ £n ~ f e 2_

7T2 " kx t .G(x)[1 + (fin

^

2

ax

) 2 ]- 1 dX

X

= f

1 (t) + f 2

(t)

We shall first deal with the second integral f 2 (t)

2 . Putting y = kx in f

2 (t) we find

00

(8) f

2 (t) = ^ ^ J

e yt G(v'y7k)[l +(~£n ^2

° 7|)

2

]

1 ^ .

The integral on the right is clearly a Laplace integral to

which we can apply the standard Abelian theorems of the

asymptotic theory of Laplace integrals [2, p, 202], We conclude that for t —>oo

(9) f

2 (

B[1 + (f/nVf)

2 ]" 1 (1/t)

p where c = [7^a/(2>/k)] and D is the constant implied by

G(vyA) = 0(y) as y

7

> 0 .

3. We now come to the much longer discussion of the integral I(t) in f^(t), i.e,

(10) I(t)

= J

e“

kx2t

[l + (fin 9

2 ]- 1 f .
-

5 -

Putting kx = (1/c )y where c = [?fa/(2>/k)] we find

oo. t

(11 ) I(t) = i I e

e [1 + Ciiny) 2 ]-1 f

=

00

7T2

e

c d (arc tan 9

Performing an integration by parts on the last integral, we

obtain

Kt) = ~[e _ ~r ~ c arctan

^

7T

a> oo __

jt * + f Te c arctan —~^d (~y)

( 12 )

7r£ 7T

>+ +

2

00 t e arctan d(~y) 7T u S C

'

O

00

We now let ty = czj noting that

J'

e "'dz = 1 , we then can

o oo write

7T*“ 7T Kt) = ip + o e ''arctan [ v^n(pz/t)

]^ 2 _

(13)

7T2

/• 00

7

2

v ’o

e ^dz + P

e zarctan[ ^ ^z

oo

o

f f

e_? [f + arctan[^^])dz.

Recalling that arctan (oo) = tt/2 and making use of an elementary trigonometric identity, we finally obtain from (13)
-6-

00

(14) I ( t ) = — C e 7arctao L

iT/ ”^)J dz.*

J o

In this integral we pot s = Lw — which yields

OO

(l£) I(t) = Q(s) ~ ~~ 2 ^

e 7!arctan i^s/Cs/n? - 1)j dz

TT2

00 e zarctan [ ?rs/( 1 - s/n?)] d:

It is this last integral which we shall evaluate for small s

4. To obtain an asymptotic representation of (15) for

s —^0 we shall prove that

( 16 ) lim

s—>0

1 °o 1 P cT 7

e a re tan [ tr s / ( 1 - s = ir t

Vi/e must show that given any > 0, however small, there

exists a ^

(£) > 0 such that

07)

00

f e 7 arctan [

ir s/(1 - $J!n z)]dz - 7f < £

for all s < d (£) . Evidently, this inequality will be satisfied if for all s < ^ (£

)

*The use of the trigonometric identity involving arctangents

was pointed out to us by P. ilenrici. Our original argument

wa s so raewh at longer.
-7-

7 \ 1 arctan [^s/(1 - s^rz)] - ' r'| dz|

( 18 ) 0

oo ®

| ^

e 7 “ arctan [?rs/(1 - s^nzOdzj + ir

| ^

e

?dzj < £

where 7 > 0 is some constant at our disposal. Since the

function

(19) h(t,z) = arctan Ort/O - t^/nz)]

is uniformly bounded for all 7 and all t > 0 and h(t ? z}

as t—^0 for z finite, we have

00 00

(20)

j e 7 ^

arctan [tts/( 1 - s /n?) ]dz

Z

< K J

e 7

d7. = Ke

Choose Z > 0 so large that max (Le v

, we

<J

) <_ — 9 Then all

we have to prove is that there exists a ^

(£) > 0 such that

( 21 ) ll e z )“ arctan l^s/O - s /nz) j - 7r |

dzj <_

for all s (£) . This, however, is now trivial since the

boundedness of h(t,z) and h ( t

,

z) -—?0 as t—>0 together

imply the bounded convergence of (21 ) to zero as s —* 0 o By virtue of (16) we can conclude from 0 5) that for

s —* 0

2

( 22 ) 9(s)

7
whence for t—* oo

(23) I(l) ~ 2~£MT/7)

Hecalling that we have from (7) and (10)

( 24 ) f

1 (t) = (~-2

- Jin ~) I(t)

7T

we finally obtain

r >^n4t k - 2^nTf a

If we compare this relation with that derived in (9) for

f^t), we see that for sufficiently large values of t the

contribution of to can neglected.

Thus, the behavior of f(t) for large values of t is

described by the asymptotic relation

5. It is possible to derive similarly higher order

terras in the asymptotic relation (25) for f^(t). In fact

we can obtain in this manner the asymptotic expansion

for t —*oo . In view of the asymptotic relation (9) for

f 2 (f) this expansion can be considered as an asymptotic

( 26 ) f ^^n4tk - 2^n~tfa
-9-

expansion, in the sense of Poincare'", of the function f(t)

for t —>oo ,

In conclusion we remark that the expansion

oo (28) 1

IT r

-kx^t

1

J^(ax)+Y^(ax) x 2 n*+tk ~ 2/nU

JLm2L

(^n^tk - 2^n ^a)

+ • » ]

can be found in the literature [1,3AJ? where it is stated

without derivation.

REFERENCES

1. Carslaw, H.5. and Jaeger, J.C. Conduct ion of Heat in

Solid s . Oxford University Pre¥s^ 19^+7.

2. Doetsch, G. Theorie und Anwendunq der Laolace-Transf or niation . (American Ed.). Dover Publications, 1 9^+X.

3. Goldstein S. Some two-dimensional Diffusion Problems with

Circular Symmetry . Proc. London Math. Soc., vol.

3^0932), pp. 51-88.

h. Jaeger, J. C. Heat Flow in the Pieoion bounded Internally

b y a Circular Cyl inder . Proc. Roy. Soc. Edinburgh

(A), vol. 61 (1942), pp. 223-228.

The American University

THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.00). Information on calibration services and fees can be

found in NBS Circular 483, Testing by the National Bureau of Standards (25 cents). Both are

available from the Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of Standards, Washington 25, D. C.
r