The author acknowledges the assistance of Geraldine Coombs, Betty Kahn, Marilyn Kemp, Betty Ruhlman, and Anna Lee Samuels for checking formulas and developing numerical examples, only a portion of which could be accommodated here.
Midwest Research Institute. (Prepared under contract with the National Bureau of Standards.)
479
11. Integrals of Bessel Functions Mathematical Properties
11.1. Simple Integrals of Beesel Functions
p , ( t ) d t
11.1.1
(Wb+v+l)>O)
11.1.3 So'J,.(t)dl=S'J.(t)dt-2 0 % k=O Jzs+l(z)
11.1.4 So'J2m+1(t)dt=1-J0(z)-2 5 k=1 J&)
Recurrence Relations
11.1.8 1 F0 ( t ) d t = ;cz,(z) +z fl{ -LO(z)Z1(z) +Ll(Z)Z"(d 1
.Z,(z) =AI, (2) + Be'**K, (z) ,v = 0'1
A and B are constants.
E&) and L,(z) are Struve functions (see chapter 12).
11.1.9
y (Euler's constant)=.57721 56649 . . .
In this and all other integrals of 11.1, z is real and positive although all the results remain valid for extended portions of the complex plane unless stated to the contrary.
11.1.23 2 Approximation 11.1.16 is from A. J. M. Hitchcock.
Polynomial approximations to Bessel functions of order zero and one and to related functions, Math. Tables Aids Comp. 11, 86-88 (1957) (with permission).
INTEQRALS OF BESSEL FUNCTION13 482 Aeymptotic Expamiom
INTEGRALS OF BESSEL FUNCTIONS 483 Recurrence Relatiom
11.2.5
11.2.6
mim
11.2.10
(9?220, w>o, wz>o, r=O) 11.2.11
Ki,(z) =-
(WZ20, Wr>O)
11.2.12
U ij r(r+3) - r (3) (r+ 1)
11.2.13 KL+,(O) -
11.2.14
rKir+l(z) = -zKir(z) + (T- l)Kir-l(z) +ZKi,-a(z)
11.3. Reduction Formulas for Indefinite Integrals
Let
11.3.1 g,,,.(z)=f’ e-P‘trZ,(t)dt
where Z&) represents any of the Bessel functions of the first three kinds or the modified Beasel functions. The parameters a and b appearing in the reduction formulae are associated with the particular type of Bessel function as delineated in the following table.
11.3.7 g.,,(z)=- 2v+l
i + 2 ” - ~ v - - 1) r
11.3.11
11.3.12
484 INTEQRALS OF BESSEL FUNCTIONS
11.3.13
11.3.24
s,’ t.Y,-,(t)dt=z.Y, (z)+- 2’r(v) (L@v>O) U
1 11.3.26 s,’ t-’l,+l(t)dt=z-’l,(z) -2,r (v+l)
11.3.27
s,’ t.~.-,(t)dt=--z.~,(2)+2’-1r(~) (gV>o)
11.3.28
I n d a t e Integrab of Productcl of h l FUIME~~OM
Let Wp(z) and 9,(z) denote any two cylinder
1- t - .K,+, (t) dt = 2 - .K.(Z)
functions of orders p and v respectively.
11.3.29
11.3.33
11.3.35
= 2 Ji(Z) k-n+l
*see page XI.
485 INTEGRALS OF BESSEL FUNCTIONS
m > o , *>-I) 11.3.41
11.4. Definite Integrals
Orthogonality Pmpertiea of -1 Functions
Let %(z) be a cylinder function of order u. In particular, let
11.4.1 %(z) -AJ,(z) +BP,(z)
where A and B are real constants. Then
11.4.2
provided the following two conditions hold: 1. A,,isarealzeroof
11.4.3 h,AV*,(Ab)-hl%(hb) =o
2. There must exist numbers kl and 4 (both not zero) so that for all n
11.4.4 klAnV*i (La) -B%(AaU) =O
In connection with these formulae, see 11.3.29. If a=O, the above is,valid provided B=O. This case is covered by the following result.
11.4.5
So'tJ,(%t)J,(%t)dt==O (mZn, u>-1)
=3[J;(41' (m=n, b=O, Y>-I)
(m=n, b ZO, Y 2 - 1)
( ~ 1 , a¶, . . . are the positive zeros of ccJ,(z)+bz..Z(z)=O, where a and b are red con- stants.
11.4.6
Definite Inregrab Over a Finite Range
11.4.7
11.4.8 Lr J0(22 sin t) cos 2ntdt=~J:(z)
11.4.9 l' Y0(2z sin t) cos 2ntdt=i Jn(z)Y,(z)
11.4.10
f J2=(2Z sin t)dt=2 R s,(z)
I
11.4.11
f J,(z sin* t)J,(z cos4 t) c8c 2tdt
486 INTEGRALS OF BESSEL FUNCTIONS
Infinite Integrals
Inyrab of the Form
11.4.12
(gP<;t a(r+v)>o) 11.4.13
pCI<;t W(r+v)>O) 11.4.14
11.4.15
(I 4<1) arc sinh b (l+b')+
SO0 sin bt K,(t)dt=
11.4.17 J,(t)dt=l (Bv>-1)
11.4.18
11.4.19
2 r
11.4.21 JmYo (t)dt =O
11.4622
11.4.23
= 0 (@'> 1)
where T,,(w) is the Chebyshev polynomial of the first kind (see chapter 22).
11.4.25
$-;t-1e-"'Jn (t)dt
2i n -- (-i)"(l-~Z)+u~~~(@)(~*<l)
=0(wZ>1)
where UJw) is the Chebyshev polynomial of the second kind (see chapter 22).
11.4.26
wbere r(a, z) is the incomplete gamma function (see chapter 6).
Integra& of the Form c."WZ,[&)dt L= 11.4.28
(B(CI+v)>O, ga'>O)
where the notation M(a, b, z) stands for the con- fluent hypergeometric function (see chapter 13).
11.4.29
(Wu>-l, @a*>O)
487 11.4.31 11.4.30
hl v-
d - 2u [I. (2) tan a w
e-""Y,,(bt)dt=-- e
+: K, (5) sec m] (19ul<r 1
11.4.31
(Wu>- 1,9a*>O) 11.4.32
Weber-Schafheitlin Type Integrals 11.4.33
sin [ p arc sin :] som J,(at) sin bt dt= (a2- b2)+ (O,<b<a)
b'r (u-p)
488 INTEGRALS OF BESSEL FUNCTIONS
Hankel-Nicholson Tope Integrda I 11.4.47
11.4.44
(a>o, Bz>O,-l<BY<29P+;)
11.4.45 OD J,(at)dt !r =- [I,(az)-L,(az)]
t’(t*+ZZ) 22.+1 (a>o, B2>0, .%>-ij 5,
b 11.4.46
(a>O, BZ>O, 9?v>-1)
11.449
Numerical Methods
11.5. Use and Extension of the Tables
For moderate values of 2, use 11.1.2 and 11.1.7- 11.1.10 as appropriate. For z suf6ciently large, use the asymptotic expansions or the polynomial approximations 11.1.11-11.1.18.
Example 1. Compute l’M Jo(t)dt to 5D.
Using 11.1.2 and interpolating in Tables 9.1 and 9.2, we have
This value is readily checked using 2 ~ 3 . 1 and h=-.05. Now IJo(z)l 51 for all 2 and IJ,,(z)l <2-4, rill for all 2. In Table 11.1, we can always choose Ihl5.05. Thus if all terms of O(h4) and higher are neglected, then a bound for the absolute error is 2+h4/48<.3.10-6 for all z if Ihl - <.05. Similarly, the absolute error for quadratic interpolation does not exceed
h3 (24 + 2)/24 < .2 - lo--?
Example 3. Interpolation of Simpson’s rule. We have
INTEQRALS OF BESSEL FUNCTION8 489 which is correct to 10D. The above procedure gives high accuracy though it may be necessary to
interpolate twice in Jo(z) to compute J o 1:+-
and Jo(r+h). A similar technique based on the trapezoidal rule is less accurate, but at most only one interpolation of Jo(z) is required.
Example 4. Cornputel Jo(t)dt a n d l YdtW
to 5D using the representation in terms of Struve functions and the tables in chapters 9 and 12.
( 3
For 2=3, from Tablea 9.1 and 12.1 Jo=-. 260052 J1= ,339059 Yo= .376850 Y1= .324674 I&= .574306 &=l. 020110
Using 11.1.7, we have
Jo(t)dt=3(-.260052) +? [(.574306)(.339059)
- (1.0201 10) (- .26OO52) 1 = 1.38757
Similarly,
Using 11.1.8 and Tables 9.8 and 12.1, one can
compute1 Io(t)dt a n d l Ko(t)dt.
Jm Jo(;)dt,S," Yo(i)dt,1 Vo( t ) - 1 l d l , l Ko(t)dt
For moderate values of z, use 11.1.19-11.1.23. For z sufficiently large, use the asymptotic ex- pansions or the polynomial approximations 11.1.24-11.1.31.
t t
Repeated Integrals of J&)
For moderate values of z and r, use 11.2.4. If r=l, see Example 1. For moderate values of 2, use tha recurrence formula 11.2.5. If 1: is large and z ~ r , see tho discussion below.
Compute j r , &)=jr(z) to 5D for 2=2 and r=0(1)5 using 11.2.6. We have
Example 5.
d r+ l (z) =zfrCz) - (r-1)jr-i (r) +S fr-,(z)
j-1(1:> = - Jl(4, f o b ) = JO(z1, f~(z> =r Jo (t) dt 0
and the termson this last line are tabulated. Thus for 2=2,
f-l=--.57672 48,fo=.22389 08,fl=1.42577 03
The recurrence formula gives
Similarly, fi= 2 (f1 4-f-1) = 1.69809 10
f3=1.20909 66,f4=.62451 73,f5=.25448 17
When r>>r, it is convenient to use the auxil- iary function
Si(%) = (T- l) !~- '+'fr(;~)
This satisfies the recurrence relation
a2gr+i(x)=$gr- (T- l)2gr-1(z) + (r- 1) (~-2)gr-2(2) r 1 3
93 (4 = Vg2 (4 - 91 (4 + Z J O O ) 1/39 91 (4 = s' JON dt, 92 (4 = 91 (3) - J1(4
0
Example 6. Compute g,(z) to 5D for z=10 and r=0(1)6. We have for z=lO,
Jo=-.24593 58, J17.04347 27, g1=1.06701 13
Thus g2331.02353 86, g3=.98827 49
and the forward recurrence formula gives
g4=.96867 36, g,=.94114 12, g,=.90474 64
For tables of 2-77(1:), see [11.16].
Repeated Integrals of &(x)
For moderate values of z, use the recurrence
Example 7. Compute Ki,(z) to 5D for 2=2 formula 11.2.14 for all T.
and r=0(1)5. We have
rKir+1(z)=-2Kir(z)+ (r-1)Kir-1(z)+~Kir-2(z) m
KLl(z) =Kl(z), Kio(z) =KO(%), Ki,(z)=S Ko(t)dt
and the functions on this last line are tabulated Thus for ,2=2,
K0=.11389 39, K1=.13986 59, Kil=.09712 06
Ki2=-2Kil+2K1=.08549 06 and
Similarly,
Kia=.07696 36, Ki4=.07043 17, Kis=.06525 22
If x/r is not large the formula can still be used provided that the starting values are sufliciently accurate to offset the growth of rounding error.
For tables of Kit(%), see [11.11].
490 INTEGRALS QF BESSEL FUNCTIONS
Now
fo(d =SK0(t)dt,j1(2) 0 = [l -zK1(41/z
the latter following from 11.3.27 with v= l . In 11.3.5, put a=l , b=-1, p=O and v=O. Let p=m. Then
Similarly starting with jl, we can compute j3 and js. If m>z, employ the recurrence formula in
backward form and write
fm-2(z>=[22fm(2)+~~~(~>+~(m--1)Ko(~)l/(m--1)2
In the latter expression, replace j,,, by gn. Take r>m and assume gr=O. gr-4, etc. Then
Fix 2. Compute gr-2,
lim gr-w(z)=j,,,(z), m=r-2k r-m
Apart from roundaff error, the value of r needed to achieve a stated accuracy for given x and m can be determined a priori. Let
Then
Qr-2k= (~-1)~(r-3)~. . . (r-2k+1)2
Qr I [ ~ K I (5) +dr- 1 )Ko(z) I/(r- 1)
since for 2 fixed, jr(Z) is positive and decreases as r increases.
Example 9. Compute j,,,(z) to 5D for 2=3 and m=0(2)10. We have
&=.03473 95 K1=.04015 64
If r= 16, ~6<.86*1O-’ q0<1.4-1O-~
Taking g16=0, we compute the following values of g14) g12, . . ., go by recurrence. Also recorded are the required values off,,, to 5D.
m
14 12 10 8 6 4 2 0
9.,
.00855 42
.01061 09
.01325 05
.01751 39
.02548 09
.04447 31
. 11936 90 1. 53994 71
.01325
.01751
.02548
.04447
. 11937 1.53995
For tables of j,,,(z), see [11.21].
References Tests
[11.1] H. Bateman and R. C. Archibald, A guide to tables of Bessel functions, Math. Tables Aids a m p . 1, 247-252 (1943). See also Supplements I, 11, IV, same journal, 1,403-404 (1943); 2,59 (1946); 2, 190 (1946), respectively.
(11.21 A. Erd6lyi et al., Higher transcendental functions, vol. 2, ch. 7 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953).
[11.3] A. ErdQyi et al., Tables of integral transforms, vola. 1, 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1954).
[11.4] W. Grirbner and N. Hofreiter, Integraltafel, I1 Teil (Springer-Verlag, Wien and Innsbruck, Austria,
[11.5] L. V. King, On the convection of heat from small cylinders in a stream of fluid, Trans. Roy. Soo. London 214A, 373-432 (1914).
[11.6] Y. L. Luke, Some notes on integrals involving Bessel functions, J. Math. Phys. 29, 27-30
1949-1950).
(1960).
fn
[11.7] Y. L. Luke, An associated Bessel function, J. Math.
[11.8] F. Oberhettinger, On some expansions for Bessel integral functions, J. Research NBS 59, 197-201 (1957) RP 2786.
[11.9] G. Petiau, La thQrie des fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, France, 1955).
[11.10] G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958).
Phys. 31, 131-138 (1952).
Tables
[11.11] W. G. Bickley ana J. Nayler, A short table of the functions Kin@), from n= l to n=16. Philos.
Mag. 7,20,343-347 (1935). Kil(z)=LmKo(t)&,
Ki,(z) =Jm Ki,,-I(t)&, n=1(1)16, z=0(.06).2
(.1)2, 3, QD.
INTEGRALS OF BESSEL FUNCTIONS 491 [11.12] V. R. Bursian and V. Pock, Table of the functions
0 (.1)16, 7D. [11.13] E. A. Chistova, Tablitay funktsii Besselya ot
deistvitel’ nogo argumenta i integralov ot nikh (Izdat. Akad. Nauk SSSR., MOSCOW, U.S.S.R.,
n=O, 1; z=0(.001)15(.01)100, 7 0 . Also tabulated are auxiliary expressions to facil- itate interpolation near the origin.
[11.14] A. J. M. Hitchcock, Polynomial approximations to Bessel functions of order zero and one and to related functions, Math. Tables Aids a m p . 11, 86-88 (1957). Polynomial approximations
for i‘Jo(t)dt and KKo(t)dt.
[ l l . l5] C. W. Horton, A short table’ of Struve functions and of some integrals involving Bessel and Struve functions, J. Math. Phys. 29, 66-68
4D, where E,@) is Struve’s function; see chapter 12.
[ l l . l6] J. C. Jaeger, Repeated integrals of Bessel functions and the theory of transienta in filter circuita, J. Math. Phys. 27, 210-219 (1948). f~(z)=
l’ Jo(t)dl, f,(z)=Jzf,l(t)dt, 2-%(z), r=1(1)7,
Z=0(1)24, 8D. Also +,(z) =~mJ~[2(~t)“]Jn(t)~p
an@), *i(z), n=1(1)7, ~=0(1)24, 4D. [11.17] L. N. Karmaeina and E. A. Chistova, Tablitay
funktaii Besselya ot mnimogo argumenta i integralov ot nikh (Izdat. Akad. Nauk SSSR., MOSCOW, U.S.S.R., 1958). e-+Zo(z), e-+Zl(z),
e=Ko(z), ezK1(z), e=, e-=f Zo(t)dl, e = c Ko(t)dt,
z=0(.001)5(.005)15(.01)100, 7D except for e’ which is 78. Also tabulated are auxiliary expres- sions to facilitate interpolation near the origin.
[11.18] H. L. Knudsen, Bidrag til teorien for antenne- systemer med he1 eller delvis rotations-symmetri. I (Kommission Has Teknisk Forlag, Copenhagen,
Denmark, 1953). J” J,(t)dt, n=0(1)8, z=
0(.01)10, SD. Also J.(t)ecodl, a=t, a=z-t.
(11.191 Y. 5. Luke and D. Ufford, Tables of the function
[11.20] C. Mack and M. Castle, Tables of Zo(z)& and
JamKo(z)&, Roy. Soc. Unpublished Math. Table
File No. 6. a=0(.02)2(.1)4, QD. [11.21] G. M. Muller, Table of the function
s,”
Kj,(z) =z-ns,’unKo(u)du,
Office of Technical Services, U.S. Department of Commerce, Washington, D.C. (1954). n=0(1)31, ~=0(.01)2(.02)5, 89.
(11.221 National Bureau of Standards, Tables of functions and zeros of functions, Applied Math. Series 37 (U.S. Government Printing Office, Washington,
![Coulomb and Bessel Functions of Complex Arguments and … · COMPLEX COULOMB AND BESSEL FUNCTIONS ... and the Fourier-Bessel equation [13] and many others, e.g., ... and gave tables](https://static.documents.pub/doc/80x56/5af3f2587f8b9a92718cd9fc/coulomb-and-bessel-functions-of-complex-arguments-and-coulomb-and-bessel-functions.jpg)
![Some Relations Between Bessel and Legendre Functions[1]](https://static.documents.pub/doc/80x56/577cd58d1a28ab9e789b14f2/some-relations-between-bessel-and-legendre-functions1.jpg)
