11. Integrals of Bessel Functions
YUDELLL. LUKE’
Contents
Page
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480
480
482
483
485
488
4aa
490
492
Table 11.2 Integrals of Bessd Functions . . . . . . . . . . . . .
494
Mathematical Properties . . . . . . . . . . . . .
11.1. Simple Istegrals of Bessel Functions . . . .
11.2. Repeated Integrals of J,,(z) and &(z) - . . .
11.3. Reduction Formulaa for Indefinite Integrals
11.4. Definite Integrals . . . . . . . . . . . .
Numerical Methoda . . . . . . . . . . . . . . .
11.5. Use and Extension of the Tables . . . . .
References . . . . . . . . . . . . . . . . . . .
Table 11.1 Integrals of Bessel Functions . . . . . .
. .
. .
. .
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. .
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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
11.1.8
F 0 ( t ) d t =;cz,(z)
p,(t)dt
+z1
fl{
-LO(z)Z1(z)
+Ll(Z)Z"(d1
.Z,(z)=AI, (2)+Be'**K,(z),v=0'1
11.1.1
A and B are constants.
E&) and L,(z) are Struve functions (see
chapter 12).
11.1.9
(Wb+v+l)>O)
y
(Euler's constant)=.57721 56649 . . .
11.1.3
So'J,.(t)dl=S'J.(t)dt-2
0
%
k=O Jzs+l(z)
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.
11.1.4
So'J2m+1(t)dt=1-J0(z)-2
5 J&)
11.1.10
Recurrence Relations
k=1
l-f=
s':
f
&(t)dt=i
11.1.11
0
Jo(t)dt+i
Y,(t)dt
481
INTEGRALS OF BE8SEL FUNCMONB
11.1.14 2'e-z~Z0(t)dt-(2r)-*
2a$-*
11.1.18
k-0
d8s.O' ~ o ( tdt) =
where the at are defined as in 11.1.12.
11.1.15 dezl"Ko(t)dt-(g) r t
OD
k
0
1
2
3
4
5
6
Polynomial Approximatione
f
81x1..
"
11.1.19
1
7
.00100 89872
.OOO63 66169
.OOO39 92825
.Ooo27 55037
.OOO12 70039
.00002 68482
2
3
4
5
6
7
11.1.17
.00178 70944
.OOO67 40148
.OOO41 00676
.Ooo25 43955
.OOO11 07299
.00002 26238
For #(z), see 6.3.
11.1.20
8 9 5 -
a9e-'JZ lo(t) dt=$ -0 dt(x/8)-t+c(x)
0
I 11.1.21
le(z)l52XlO-'
k
0
1
2
3
4
5
6
dr
,39894
.03117
.00591
.00559
--.01148
.01774
-. 00739
23
34
91
56
58
40
95
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).
2
(-)*et (x/7)-t+ t (z)
le(~)l
52x10-7
(-)'aF'
where the at are defined m in 11.1.12.
P
6
k-0
k-0
11.1.16
71x5..
I 11.1.22
11.1.23
ek
1.25331
0.11190
.02576
.00933
.00417
.00163
.00033
414
289
646
994
454
271
934
INTEQRALS OF BESSEL FUNCTION13
482
Aeymptotic Expamiom
11.1.30
45x5
Ic(z)I 56X lo-'
where
k
0
1
2
3
4
5
6
11.1.25
11.1.31
1.25331
0.50913
.32191
.26214
.20601
.11103
.02724
41
39
84
46
26
96
00
5 5 2 5 Q)
where
13
co=l, c1=8
11.1.27
k
0
1
2
3
4
5
6
7
8
9
10
where c k is defined as in ll.l.!Z?.
Polynomial Approximations
11.1.29
55x5
fk
0.39893
. 13320
-. 04938
1.47800
-8.65560
28.12214
-48.05241
40.39473
-11.90943
-3.51950
2.19454
14
55
43
44
13
78
15
40
95
09
64
Repeated Intqgrals of J,(z)
k
bk
0
1
2
3
4
5
6
7
8
9
1. 0
0. 15999 2815
. 10161 9385
. 13081 1585
.20740 4022
.28330 0508
.27902 9488
. 17891 5710
.06622 8328
.01070 2234
1. 0
0.81998 6629
11.2.3
10702 2336
483
INTEGRALS OF BESSEL FUNCTIONS
11.2.5
Recurrence Relatiom
11.2.6
mim
11.2.10
(9?220,
11.2.11
w>o,
wz>o, r=O)
11.3.7 g.,,(z)=-
2v+l
Ki,(z) =-
(WZ20, Wr>O)
11.2.12
U
11.2.13 KL+,(O)-
ij r(r+3)
-r (3) (r+ 1)
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
i
+ 2 ” - ~ v -1)- r
11.3.11
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.
11.3.12
484
INTEQRALS OF BESSEL FUNCTIONS
11.3.13
11.3.26
s,’ t-’l,+l(t)dt=z-’l,(z)-2,r
1
(v+l)
11.3.27
s,’t.~.-,(t)dt=--z.~,(2)+2’-1r(~)
11.3.28
(gV>o)
1-
t -.K,+,(t)dt =2- .K.(Z)
I n d a t e Integrab of Productcl of h
l FUIME~~OM
Let Wp(z) and 9,(z) denote any two cylinder
functions of orders p and v respectively.
11.3.29
11.3.33
11.3.35
11.3.24
s,’t.Y,-,(t)dt=z.Y,
(z)+-2’r(v)
U
=
(L@v>O)
2 Ji(Z)
k-n+l
*seepage XI.
INTEGRALS OF BESSEL FUNCTIONS
485
2. There must exist numbers kl and
not zero) so that for all n
11.4.4
klAnV*i (La)-B%(AaU)
4
(both
=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,
)
I
-
>
Y
(m=n, b ZO, 2 - 1)
Y
. . . are the positive zeros of
ccJ,(z)+bz..Z(z)=O, where a and b are red constants.
( ~ 1 ,a¶,
11.4.6
m>o,
11.3.41
*>-I)
Definite Inregrab Over a Finite Range
11.4. Definite Integrals
Orthogonality Pmpertiea of1-
Let %(z) be a cylinder function of order u.
In particular, let
f
11.4.7
Functions
11.4.8
J2=(2Z sin t)dt=2
R s,(z)
Lr
l'
J0(22sin t) cos 2ntdt=~J:(z)
I
11.4.1
%(z) -AJ,(z)
+BP,(z)
where A and B are real constants. Then
11.4.2
11.4.9
Y0(2zsin t) cos 2ntdt=i Jn(z)Y,(z)
11.4.10
11.4.11
provided the following two conditions hold:
1. A,,isarealzeroof
11.4.3
h,AV*,(Ab)-hl%(hb)
=o
f J,(z sin*t)J,(z cos4 t) c8c 2tdt
486
INTEGRALS OF BESSEL FUNCTIONS
Infinite Integrals
11.4.23
I n y r a b of the Form
11.4.12
=0 (@'> 1)
where T,,(w) is the Chebyshev polynomial of the
first kind (see chapter 22).
(gP<;t
a(r+v)>o)
11.4.25
11.4.13
$-;t-1e-"'Jn
pCI<;t
W(r+v)>O)
11.4.14
11.4.17
(-i)"(l-~Z)+u~~~(@)(~*<l)
=0(wZ>1)
where UJw) is the Chebyshev polynomial of the
second kind (see chapter 22).
11.4.26
11.4.15
SO0
--2in
(t)dt
sin bt K,(t)dt=
arc sinh b
(l+b')+
J,(t)dt=l
(I 41<)
(Bv>-1)
wbere r(a,z) is the incomplete gamma function
(see chapter 6).
11.4.18
Integra& of the Form
11.4.28
11.4.19
2
r
L=
c."WZ,[&)dt
(B(CI+v)>O,
ga'>O)
where the notation M(a, b, z) stands for the confluent hypergeometric function (see chapter 13).
11.4.21
JmYo(t)dt=O
11.4.29
11.4622
(Wu>-l,
@a*>O)
487
11.4.30
11.4.31
e-""Y,,(bt)dt=--
+: K,
d e
2u
(5)
sec m]
h
v-l
[I.(2)
tan
(19ul<r1
a
w
11.4.31
11.4.32
(Wu>-
1,9a*>O)
som
J,(at) sin bt dt=
:]
sin [ p arc sin
(a2-b2)+
Weber-Schafheitlin Type Integrals
11.4.33
b'r (u-p)
(O,<b<a)
488
INTEGRALS OF BESSEL FUNCTIONS
Hankel-Nicholson Tope Integrda
I
11.4.44
(a>o,
11.4.47
Bz>O,-l<BY<29P+;)
11.4.45
b
OD
J,(at)dt =-
t’(t*+ZZ)
(a>O, BZ>O,
!r
[I,(az)-L,(az)]
22.+1
(a>o, B2>0,
.%>-ij5,
9?v>-1)
11.449
11.4.46
Numerical Methods
11.5. Use and Extension of the Tables
For moderate values of 2, use 11.1.2 and 11.1.711.1.10 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 . 1and
h=-.05. Now IJo(z)l51 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’srule. We have
13.M
Jo(t)dt=2[.32019 09 +.31783 69 +.04611 52
+. 00283 19 +.00009 72+ .OOOOO 211
= 1.37415
Example 2. Compute
l.’’
Jo(t)dt to 5D by
interpolation of Table 11.1 using Taylor’sformula.
We have
h‘
+3[Jz(z)-Jo(z)I+gg [3J1(4--Ja(dI+
ha
and with IhII .05, it follows that
- -
lRl<.9
10-10
Then with z=3.0 and h=.05,
Thus if 2=3.0 and h=.05
~~J0(t)dt=1.387567+(.05)(-.260052)
-(.00125)(.339059)
+ (.OOOOlO) (.746143) = 1.37415
~‘06Jo(t)dt=1.38756
72520+-( 05) [-.26005 19549
6
+4(-.26841 13883)-.27653 495991
=1.37414 86481
489
INTEQRALS OF BESSEL FUNCTION8
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
(
3
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.
The recurrence formula gives
fi=2 (f1
Similarly,
Using 11.1.7, we have
Jo(t)dt=3(-.260052)
= 1.38757
+?
[(.574306)(.339059)
-(1.020110)(- .26OO52)1
= 1.69809 10
f3=1.20909 66,f4=.62451 73,f5=.25448 17
When r>>r,
iary function
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.
For 2=3, from Tablea 9.1 and 12.1
J1= ,339059
Jo=-. 260052
Yo= .376850
Y1= .324674
I&=
.574306
&=l. 020110
4-f-1)
it is convenient to use the auxil-
Si(%)= (T- l)!~-'+'fr(;~)
This satisfies the recurrence relation
a2gr+i(x)=$gr91 (4=
s'
0
(T-
l)2gr-1(z)
+ (r- 1) (~-2)gr-2(2) r 1 3
JONdt, 92(4=91(3)-J1(4
93
(4= Vg2(4-91(4+Z J O O )
1/39
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
Similarly,
and the forward recurrence formula gives
Using 11.1.8 and Tables 9.8 and 12.1, one can
For tables of 2-77(1:), see [11.16].
compute1 Io(t)dt a n d l Ko(t)dt.
Jm
Jo(;)dt,S," Yo(i)dt,1 V o ( t ) - 1 l d l , l
t
Ko(t)dt
t
For moderate values of z, use 11.1.19-11.1.23.
For z sufficiently large, use the asymptotic expansions or the polynomial approximations
11.1.24-11.1.31.
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 ~ rsee
, tho discussion below.
Example 5. Compute j r , &)=jr(z) to 5D for
2=2 and r=0(1)5 using 11.2.6. We have
=r
(r)+S fr-,(z)
j-1(1:>
= -Jl(4, f o b ) =JO(z1, f~(z>
Jo (t)dt
0
and the termson this last line are tabulated. Thus
for 2=2,
f-l=--.57672
Repeated Integrals of &(x)
For moderate values of z, use the recurrence
formula 11.2.14 for all T.
Example 7. Compute Ki,(z) to 5D for 2=2
and r=0(1)5. We have
rKir+1(z)=-2Kir(z)+(r-1)Kir-1(z)+~Kir-2(z)
m
Repeated Integrals of J&)
d r + l(z)=zfrCz)- (r-1)jr-i
g4=.96867 36, g,=.94114 12, g,=.90474 64
48,fo=.22389 08,fl=1.42577 03
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
and
Ki2=-2Kil+2K1=.08549 06
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
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
Now
fo(d
= [l -zK1(41/z
=SK0(t)dt,j1(2)
0
Then
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
Qr-2k=
Qr
jn(4
=[(m -1>Ynl-2(4 -2KI (4
-4m-1>~0(41/2
(m>l>
Using tabular values of joand f l , one can compute
in succession j2,
js,. . . provided that m/x is not
large.
Example 8. Computej,,,(z)to 5 D for 2 = 5 and
m=0(1)6. We have, retaining two additional
decimals
&= -00369 11
Thus
jo=l. 56738 74
K1=. 00404 46
fi=. 19595 54
(~-1)~(r-3)~.. . (r-2k+1)2
I [ ~ K(5)
I+dr- 1)Ko(z)I/(r- 1)
since for 2 fixed,jr(Z) is positive and decreases as r
increases.
Example 9. Computej,,,(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.
j2=.05791 27,j,=.01458 93,j6=.00685 36
Similarly starting withjl,we can computej3andjs.
If m>z, employ the recurrence formula in
backward form and write
m
14
12
10
8
6
4
2
0
fm-2(z>=[22fm(2)+~~~(~>+~(m--1)Ko(~)l/(m--1)2
In the latter expression, replace j,,,
by gn. Fix 2.
Take r>m and assume gr=O. Compute gr-2,
gr-4, etc. Then
lim gr-w(z)=j,,,(z), m=r-2k
fn
9.,
.00855
.01061
.01325
.01751
.02548
.04447
. 11936
1. 53994
42
09
05
39
09
31
90
71
.01325
.01751
.02548
.04447
. 11937
1.53995
For tables of j,,,(z),see [11.21].
r-m
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,
1949-1950).
[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
(1960).
[11.7] Y. L. Luke, An associated Bessel function, J. Math.
Phys. 31, 131-138 (1952).
[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).
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)&,
(.1)2, 3, QD.
n=1(1)16, z=0(.06).2
491
INTEGRALS OF BESSEL FUNCTIONS
[11.12] V. R. Bursian and V. Pock, Table of the functions
Akad. Nauk,
Leningrad, Inst. Fiz. Mat.,
Trudy (Travaux) 2, 6-10 (1931). JmKo(t)dl,
L-
z=O(.1)12, 7D; e=
Zo(t)dt, z=O(.1)6,
Ko(t)dt, z=O(.l)l6, 7D;
7D; e-=
Zo(t)dt, z=
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 facilitate 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)dtand KKo(t)dt.
[ll.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
(1960). C,(z)
4D; D,(z)
=$
tJ,(t)dt, n= 1(1)4,2=O(.I)10,
=J’t”E.(t)dt,n=0(1)4, ~=0(.1)10,
4D, where E,@) is Struve’s function; see
chapter 12.
[ll.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,
[11.18] H. L. Knudsen, Bidrag til teorien for antennesystemer med he1 eller delvis rotations-symmetri.
I (Kommission Has Teknisk Forlag, Copenhagen,
Denmark, 1953).
J”J,(t)dt,
0(.01)10, SD. Also
f
n=0(1)8, z=
J.(t)ecodl, a=t,a=z-t.
(11.191 Y. 5.Luke and D. Ufford, Tables of the function
&(z) =6’Ko(t)dt.
Math. Tables Aids Comp.
UMT 129. Z(z)= -[r+h(~/2)1Ai(~)
+ Aa(z),
Ai@), A&). z=0(.01).5(.05)1, 8D.
[11.20] C. Mack and M. Castle, Tables of
s,”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
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,
D.C., 1954). (1) pp. 21-31: s,’Jo(t)dt,
z=O(.01)10, 10D.
(2) pp. 33-39:
~=0(.1)10(1)22,
10D;
Y~(t)dt,
Jo(t)dt/t,
F(z)=Jm Jo(t)dl/t
+In (z/2), z=0(.1)3, 10D; F(”(z)/n!, z=
lO(l)22, n=O(1)13, 12D.
[11.23] National Physical Laboratory, Integrals of Bessel
functions, Roy. Soc. Unpublished Math. Table
[11.24] M. Rothman, Table of
PO(Z)&
for 0(.1)20(1)26,
Quart. J. Mech. Appl. Math. 2, 212-21.7 (1949).
Z=0(1)24, 8D. Also +,(z) =~mJ~[2(~t)“]Jn(t)~p
85-98.
Jo(t)dt for large z,
[11.26] P. W. Schmidt, Tables of
an@),*i(z), n=1(1)7, ~=0(1)24, 4D.
[11.17] L. N. Karmaeina and E. A. Chistova, Tablitay
J. Math. Phys. 34, 169-172 (1955). 2=10(.2)40,
funktaii Besselya ot mnimogo argumenta i
6D.
integralov ot nikh (Izdat. Akad. Nauk SSSR.,
(11.261 G. N. Watson, A treatise on the theory of Bessel
MOSCOW,U.S.S.R., 1958). e-+Zo(z), e-+Zl(z),
functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958). Table VIII, p. 752:
e=Ko(z),ezK1(z), e=, e-=f Zo(t)dl, e = cKo(t)dt,
i p ~ ( t ) d t , ; f Y ~ ( t ) d t , 2=0(.02)1, 7D, with
z=0(.001)5(.005)15(.01)100, 7D except for e’
the first 16 maxima and minima of the integrals
which is 78. Also tabulated are auxiliary expresto 7D.
sions to facilitate interpolation near the origin.
6’