Table 9.12. Kelvin Functions-Orders 0 and 1 (O<z<5) ber z, bei z, berl 2, bei, z ker z, kei z, kerl z, keil z z=0(.1)5, IOD, 9D
9s or 10s
. . . . . .
Kelvin Functions-Auxiliary Table for Small Arguments (O<z<l). . . . . . . . . . . . . . . . . . . . . ker z+ber z In z, kei z+bei z I n z z(kerlz+berl z In z), z(kei, z+beil z I n z) z=O(.1)1, 9D
The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K. Cherwinski, Elizabeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and Ruth Zucker of the National Bureau of Standards, and N. F. Bird, C. W. Clenshaw, and Joan M. Felton of the National Physical Laboratory in the preparation and checking of the tables and graphs.
9. Bessel Functions of Integer Order
Mathematical Properties
Notation The tables in this chapter are for Bessel func-
tions of integer order; the text treats general orders. The conventions used are:
z=z+iy; z, y real. n is a positive integer or zero. v, p are unrestricted except where otherwise
indicated; v is supposed real in the sections devoted to Kelvin functions 9.9, 9.10, and 9.11.
The notation used for the Bessel functions is that of Watson [9.15] and the British Association and Royal Society Mathematical Tables. The function Y,(z) is often denoted Nv(z) by physicists and European workers.
Aldis, Airey: Other notations are those of:
G,(z) for -+rY,(z),K,(z) for (-)*K,(z).
Clifford:
C,(z) for ~-4~J,,(2fi).
Gray, Mathews and MacRobert [9.9]:
Y,(z) for +rY,(z)+ On 2--r)J,(z),
Y,(z) for revr* sec(va) Y,(z),
0, (2) for +7riH:1) (2).
-
Jahnke, Emde and Losch [9.32]:
a&) for r (Y+ 1)( $2) - ’J, (2).
Jeff reys :
H.s,(z) for HY(z), Hi,(z) for H?)(z),
Kh,(z) for (2/a)K,(z).
Heine:
K, (2) for- ~PY, (2).
Neumann:
Yn(z) for )?rY,(z)+(ln 2--y)Jn(z).
Wbittaker and Watson [9.18]:
K,(z) for cos(vir)K,(z).
358
Bessel Functions J and Y
9.1. Definitions and Elementary Properties
Differential Equation
22-+2 d2w dw -+(22-v2)w=O 9.1.1 dz2 dz
Solutions are the Bessel functions of the f i s t kind J*.(z), of the second kind Yv(z) (also called Weber’s function) and of the third kindH$”(z), H:z)(z) (also called the Hankel functions). Each is a regular (holomorphic) function of z throughout the z-plane cut along the negative real axis, and for fixed z ( f0 ) each is an entire (integral) func- tion of v. When v= &n, Jv(z) hrts no branch point and is an entire (integral) function of z.
Important features of the various solutions are as follows: Jv(z)(9?v20) is bounded as z+O in any bounded range of arg z. Jv(z) and J-,(z) are linearly independent except when v is an integer. J.(z) and Y,(z) are linearly independent for all values of v. H!’)(z) tends to zero as IzI+- in the sector
O<arg Z<T; Hi2)(z) tends to zero as lzl--+m in the sector -r<arg z<O. For all values of v, H!”(z) and H!”(z) are linearly independent.
Relatione Between Solutions
J,(z) COS (m)- J-,(z) 9.1.2 Y,(z)= sin (m)
The right of this equation is replaced by its limiting value if v is an integer or zero.
9.1.3
BESSEL FUNCTIONS OF INTEGER ORDER 359
L I
FIGURE 9.2. Jlo(z), Ylo(x), and M o (z) = JJ:o (XI + Eo (XI.
FIGURE 9.1. Jo(z), YO@), Jl(z), Yl(z>.
’ FIGURE 9.3. J,.(lO) and y”(10)-
FIGURE 9.4. Contour lines of the modulus and phase of the Hankel Function HP(x+iy)=MoefSo. From E. Jahnke, F. Emde, and F. Losch, Tables of higher functions, McGraw-Hill Book CO., Inc., New York, N.Y., 1960 (with permission).
360 BESSEL FUNCTIONS OF INTEGER ORDER
Limiting Forms for Small Arguments
When v is fixed and z+O
9.1.7 Jv(z)-($z)v/r(V+l) (VZ-1, -2, -3, . . .)
9.1.8 Yo(z)- - iH~1)(z)~~H~2)(z)~(2/~) In z
9.1.9
YJZ) - - - i ~ : l ) (2) - i ~ : 2 ) (z) - - ( I / ~ ) r (.) ($2) -I ( 9 v >O )
Ascending Series
9.1.11
(tz”>” ($E!)-” n-1 (n-k-l)! k! Y,(z)=--
T k=O
where $(n) is given by 6.3.2.
9.1.13
Integral Representations 9.1.18
1 ’ J~ (2) =; S, cos (z sin
9.1.19
~,(z)=f I” cos (z cos e) {r+h (22 sin2 e) 1 d~
9.1.20
9.1.21
COS (zsin e-&)&
9.1.22
9.1.26
In the last integral the path of integration must lie to the left of the points t=O, 1, 2, . . . .
BESSEL FUNCTIONS OF INTEGER ORDER 361
(k=O, 1,2, . . .) 9.1.31
v v+l P.+l+T,=; p,-- b PV+l
v v+ l rv+1+qv=- b p,-- a P?+l
1 1 3 8.=2 P..+l+Z P.-I-& p,
and
9.1.34 4 pa,- qvr,=- gab
Analytic Continuation
In 9.1.35 to 9.1.38, m is an integer.
9.1.35 Jv( ze’” = em- J,( z)
9.1.36
Y,(zemrf) =e-mvrfYv(z) +2i sin(mvr) cot(vr) J,(z)
9.1.37
sin (v~)H:~) (amr f, = -sin { (m- 1) v r } H;l) (z) 9.1.38 sin(vr)H;’) (amr? =sin { (m+ 1) v r ) 23:’) (z)
9.1.39
sin(mvr) H!’) (2)
+ e v r f sin (mvr) H:’) ( z)
H!l)(zerf)= -e-vrfH$’)
H:’) (ze-rf) = -p*Hil’(z) 0
9.1.40 - -
JIG) =Jv(z) Y,G) = Yv(z)
H;l)(Z)=m) Hr) (Z )=Hm (V real)
Generating Function and Amia t ed Serier
m
9.1.41 eW-W)= t *Jk(Z) ( t W k--m
m
k = l 9.1.42 cos (z sin e) =Jo(z) +2 C Jzk(z) cos (2M)
Jn(2Z)=e Jr(Z)J,-i (2) 4-2 c (-)*Jdz)J,+dZ) k-0 k-1
Graf”e
9.1.79
%‘’(W) sin vX= c W,+&)J*(v) s . ka(lveftal<lul)
Gegenbauer’s
9.1.80
-- * d ~ ) - ~ , ~ ( ~ ) 2 (v+k) Wp+ t ( ~ ) JP+$v) c(x’(cos a)
cos m cos k--m
W’ k-0 U’ V
(v#O,-l , . . ., Ive**al<IuI)
OF INTEGER ORDER 363 In 9.1.79 and 9.1.80,
w=~(u2+2?-2uv cos a), u-v cos a=w cos x, v sina=w sin x
the branches being chosen so that 2o--vu and x+O as v+O. C‘X)(cos a) is Gegenbauer’s polynomial (see chapter 22).
Ge g e n hw’ s addition theorem.
If u, v are real and positive and 0 Sa S a, then w, x are real and non-negative, and the geometrical relationship of the variables is shown in the dia- gram.
Thc restrictions Ive*‘”l< 1.1 are unnecessary in 9.1.79 when g=J and ,, is an integer or zero, and in 9.1.80 when Y=J. Degenerate Form (u= 0):
9.1.81 et0 “06a=r(v)(32))-v 2 (v+k)i*J,+r(v)C:”(cos a)
k-0 ( Y Z O , -1, . . .)
Neumann’s Expansion of an Arbitrary Function in e Series of Beasel Function8
9.1.82 f(z)=aoJo(z)+2 2 aJ&) (Izl<c) k-1
where c is the distance of the nearest singularity off(z) from z=O,
9.1.83 a*=- 1 J f(t)O*(t)dt (O<C’<C) 2a-i +e’
and O,(t) is Neumann’s polynomial. is defined by the generating function
9.1.84
The latter
L=JO(z)Odt)+2 t-2 5 k-1 J&)odt> (Izl<ltl)
O,(t) isapolynomialof degreen+l in l/t; Oo(t)=l/t,
9.1.85
(n=1,2,. . .) n(n-k-l)! 2 “-a+* w - 4 -’% klo k! (t>
The more general form of expansion
j(z) =ao~.(z> +2 5 aJv+*(z) 9.1.86 k-1
364 BESSEL F"CT1ONS OF INTEGER ORDER
also called a Neumann expansion, is investigated in [9.7] and [9.15] together with further generaliza- tions. Examples of Neumann expansions are 9.1.41 to 9.1.48 and the Addition Theorems. Other examples are
9.1.87
(VZO, -1,-2,. . .)
9.1.88 n!($z)-" n-1 (+Z>Vk(Z) Y, (2) = - -
T (n-k)k!
where +(n) is given by 6.3.2.
9.1.89
9.2. Asymptotic Expansions for Large Arguments
Principal Asymptotic Forms
When Y is fixed and lz1+co
9.2.2
Yl(4 =m
Hankel's Asymptotic Expansions
When v is k e d and 1zI+-
9.2.5
J,(z)=J-a/(rz){P(v, 2) cosx-QQ(v, 2) sinx)
( I arg 4 <*I
9.2.6 Yv(z)=~'2/(rz){P(v, z) sin x+Q(v, z) cosx}
(la% 21 < 9.2.7 H,'"(z)=,/G){P(v, z)+iQ(v, z)jefx
(-T<arg 2<2r)
9.2.8
H!2) (z) = d m { P( v, z) -iQ( v, z) }e-fx
where x=z-(+++)u and, with 4v2 denoted byp,
9.2.9
(-2a<arg z<r)
. . . + b-1) (p-9) (p-25) (p-49) - 4! (82)'
9.2.10
+ . . . -cc--l (~--~)(P-9~(cC--5) 82 3! ( 8 ~ ) ~
If v is real and non-negative and z is positive, the remainder after k terms in the expansion of P(v, z) does not exceed the (k+l)th term in absolute value and is of the same sign, provided that k>$u-f. The same is true of &(v,z, provided that k>$v-t.
Asymptotic Expaneione of Derivative8
With the conditions and notation of the pre- ceding subsection
9.2.11
J L ( z > = J ~ { --~(v, z) sinx--S(v, z) cos x}
9.2.12
Y:(z) =JG) {R (v, z) cos x- S(v, z) sin x)
(la% zl<r)
(larg zl<r> 9.2.13
H;l)'(z) = 42/(~z){iR(v, z) --S(v, z)}e'x
(-7r<arg z<2r) 9.2.14
~ ! 2 ) ' ( z )= , /m{ - - i ~ ( v , z)--~(u, z))e-'X (-2r<arg z<r)
BESSEL mCTIONS OF INTEGER ORDER 365
9.2.15
9.2.16
Modulus and Phase
For real v and positive x
9.2.17
M, = IW (4 I =.\I{ cmx) + E(2) 1 8,=arg H:’)(x)=arctan { Y,(x)/Jv(x))
The corresponding expansions for (Y+ z ~ ~ / ~ ) and H!a)(v+~v1/3) are obtained by use of 9.1.3 and 9.1.4; they are valid for --)r<arg v<#r and -#?r<arg v<+r, respectively.
9.3.27
22/3 (ID hk(z) J:(v+zv~/~) --y2/3 Ai' (-2ll3z) { 1+c -) k = l V
9.3.28
hdz) 2213 Y:(v+ zV1I3) - y2/3 Bi' (-2ll3z) { 1+C k-1 -} V where
These are more powerful than the previous ex- pansions of this section, save for 9.3.31 and 9.3.32, but their coefficients are more complicated. They reduce to 9.3.31 and 9.3.32 when the argument equals the order.
9.3.35
Ai'(v2I3{) bk({) + v5/3 2 7 1
9.3.36
IF INTEGER ORDER
9.3.37
e2ri13Ai I (e2ri13v2f3 bd l ) + v5/3 %-I k=O V*
When v++ m , these expansions hold uniformly with respect to z in the sector larg z] 5 ?r- e, where e is an arbitrary positive number. The corre- sponding expansion for H?'(vz) is obtained by changing the sign of i in 9.3.37.
Here
9.3.38
equivalently,
9.3.39
the branches being chosen so that { is real when z is positive. The coefficients are given by
9.3.41 6s f l
9 p,=-- A, (284-1) (2~+3) . . . (68-1)
s! (144)' 6s- 1 ha=
Thus a,,({) = 1,
9.3.42 5 1 5 1
5 1 5 1
bo({) =-=+? 124(1 -z2)3/2-8(1 - z2)i 1
1 =-- 48l2+(-s)i '24(z2- 1)312+8(~2- 1))
Tables of the early coefficients are given below. For more extensive tables of the coefficients and for bounds on the remainder terms in 9.3.35 and 9.3.36 see t9.381.
BESSEL FUNCTIONS OF INTEGER ORDER 369
Uniform Expansions of the Derivatives
With the conditions of the preceding subsection
9.3.4!3
where
9.3.46 2kS1
8=0 ck({)=-r* C ~ ~ { - ~ ~ / ~ v 2+ ~+ 1 ( (1--2)-tl
d&) =E ia{-3sflv2k-x{ (1 -z2)-t} 2k
8 =O
and vk is given by 9.3.13 and 9.3.14. For bounds on the remainder terms in 9.3.43 and 9.3.44 see [ 9.381.
Jo(x) =x-tfo cos e, Yo(x) =x-+j, sin e, fo=.79788 456- .OOOOO 077(3/~) - .00552 740(3/~)'
-.00009 512(3/~)'+ .00137 237(3/~)'
- .00072 805(3/~)~+ .00014 476(3/~)*+t
[el< 1.6X lo-* 2 Equations 9.4.1 to 9.4.6 and 9.8.1 to 9.8.8 are taken
from E. E. Allen, Analytical approximations, Math. Tables Aids Comp. 8, 240-241 (1954), and Polynomial approxi- mations to some modified Bessel functions, Math. Tables Aids Comp. 10, 162-164 (1956) (with permission). They were checked at the National Physical Laboratory by systematic tabulation; new bounds for the errors, e, given here were obtained as a result.
For expansions of Jo(s), Yo(s>, Jl(z), and Yl(x) in series of Chebyshev polynomials for the ranges 05s<S and 0<8/z5l, see t9.371.
9.5. Zeros
Real Zeros
When Y is real, the functions J,(z), Jl(z), Y,(z) and Y:(z) each have an infinite number -of real zeros, all of which are simple with the possible exception of z=O. For non-negative Y the 6th positive zeros of these functions are denoted by
- *t J,.,~, J,.,~, yv . a and Y:,a respectively, except that z=O is counted as the h t zero of JA(z). Since Ji(z)=-Jl(z), it follows that
*I . 9.5.1 jL,i=O, ~ o . s= j i . s - i (s=2, 3, . .)
The positive zeros of any two real distinct cylinder functions of the same order are interlaced, as are the positive zeros of any real cylinder function %‘,(z), defined as in 9.1.27, and the contiguous function V,+,(Z).
9.5.3 Vp(z) = J,(z) cos(d) + Y,(z) sin(?rt)
where t is a parameter, then
If pv is a zero of the cylinder function
9.5.4 u:(P,)=u.-l(P.) = - U . + , ( P . >
If u. is a zero of W;(z) then
The parameter t may be regarded as a continuous variable and pr, u, as functions p . ( t ) , u,(t) of t. If these functions are fixed by
+ .02918~-~'~- .0068~-~+ . . . ) 9.5.21 Y&:, J m.57319 40~-"~(1- .36422 O V - ~ / ~
+.09077~-*~+.0237v-~+ . . . )
Corresponding expansions for s=2, 3 are given in [9.40]. These expansions become progressively weaker as s increases; those which follow do not suffer from this defect.
Uniform Asymptotic Expansions of Zeros and Associated Values for Large Orders
9.5.22 j,.,-vz(r)+C OD f k ( r > with {= ~ - ~ / ~ a , k= l
9.5.23
with {=~ - ~ / ~ a ,
(D
9.5.24 j;,,-vz({)+C With {=~ - ~ / ~ a :
9.5.25 b-1 P-'
where a,, a: are the sth negative zeros of Ai@), Ai'(z) (see 10.4), z=z(T) is the inverse function defined implicitly by 9.3.39, and
where bo({), co({) appear in 9.3.42 and 9.3.%. Tables of the leading coefficients follow. More ex- tensive tables are given in [9.40].
The expansions of yv. a, YXyv, a), y:. I and Y.(Y:. J corresponding to 9.5.22 to 9.5.25 are obtained by changing the symbols j, J, Ai, Ai', a, and a: to y, Y, -Bi, -B?, b, and b: respectively.
372
h (i-)
1. 25992
BESSEL FUNCTIONS OF INTEQER ORDER
fl (i-)
0. 0143 -. ~ _ . .
1.22076 1. 18337 1. 14780 1. 11409 1.08220
-. 0142 .0139 .0135 .0131
0.0126
0. 40 .35 .30 .25 .20
1.528915 1. 541532 1. 551741 1.559490 1.564907
0. 15 . 10 .05 . 00
1.568285 1.570048 1. 570703 1.570796
-i-
1.000000 1. 166284 1.347557 1.543615
-0.007 -. 005 -. 004 -. 003 -. 003
-0.002
-0. 1260 -. 1335 -. 1399 -. 1453 -. 1498
-0. 1533
-0.010 -. 010 -. 009 -. 009 -. 008
-0. 008
0. 000 .002 .004 .005 .006
0.006
0. 0 0. 2 0. 4 0. 6 0. 8 1. 0
-r - 1. 0 1. 2 1. 4 1. 6 1. 8
2. 0 2. 2 2. 4 2. 6 2. 8
3. 0 3. 2 3. 4 3. 6 3. 8
4.0 4.2 4.4 4.6 4.8
5. 0 5. 2 5. 4 5. 6 5. 8
6. 0 6. 2 6. 4 6. 6 6. 8
7. 0
1.754187 1.978963
1.978963 2. 217607 2. 469770 2. 735103
0.0120 .0121 .0115 . 0110 .0105
0.0100 .0095 .0091 .0086 .0082
0.0078 .0075 .0071 .0068 .0065
0.0062 .0060 .0057 .0055 .0052
0.0050 .0048 .0047 .0045 .0043
0.0042 .0040 .0039 .0037 .0036
0.0035
-0.002 -. 002 -. 001 -. 001 -. 001
-0.001 -0.001
-0.008 -. 004 -. 002 -. 001 -. 001
-0.001
0.006 .004 .003 .002 .002
0.001 .001 . 001 .001
0.001
1.08220 1. 05208 1.02367 0.99687 .97159
0.94775 . 92524 .90397 .a8387 . 86484
0.84681 .a2972 . 81348 . 79806 .78338
0. 76939 .75605 . 74332 . 73115 .71951
0. 70836 .69768 . 68742 . 67758 . 66811
0. 65901 .65024 .64180 .63366 .62580
0.61821
-0. 1533 -. 1301 -. 1130 -. 0998 -. 0893 3.013256
3.303889 3. 606673 3. 921292 4. 247441 4.584833
4. 933192
-0.0807 I -. 0734 -. 0673 -. 0619 -. 0573
- I -0.0533 -. 0497 -. 0464 -. 0436 -. 0410
-0.0386 -. 0365 -. 0345 -. 0328 -. 0311
-0.0296 -. 0282 -. 0270 -. 0258 -. 0246
5. 292257 5.661780 6.041525 6. 431269
6. 830800 7. 239917 7. 658427 8.086150 8. 522912
8.968548 9.422900 9.885820
10.357162 10.836791
11.324575 11. 820388
-0.0236 -. 0227 -. 0218 12. 324111
12. 835627 13.354826
13.881601
-. 0209 -. 0201
-0.0194
Complex Zeros of J,(s)
When v> -1 the zeros of J,(z) are all real. If v<-1 and v is not an integer the number of com- plex zeros of J,(z) is twice the integer part of (-v); if the integer part of (-v) is odd two of these zeros lie on the imaginary axis.
If v20, all zeros of JL(z) are real.
( --f)W)
1.62026 1.65351 1. 68067 1. 70146 1.71607
1.72523 1.73002 1.73180 1. 73205
81(3)
-0.0224 -. 0158 -. 0104 -. 0062 -. 0033
-0.0014 -. 0004 -. 0001 -. 0000
0.0040 .0029 .0020 . 0012 .0006
0.0003 . 0001 .moo .woo
Complex &roo of Y,(r)
When vis real the pattern of the complex zeros of P,(z) and Yv(z) depends on the non-integer part of v. Attention is confined here to the case u=n, a positive integer or zero.
a=m= . 6 6 2 7 4 . . .
b = + J m I n 2=.19146 . . . and b=1.19968 . . . is the positive root of coth t =t. There are n zeros near each of these curves. Asymptotic expansions of these zeros for large n
I
FIGURE 9.6. Zeros ofHi’)(z) and Hi”’(z) . . . larg zl<?r.
The asymptote Of the solitary infinite curve is given bY
Y~=-+In2=-.34657 . . .
BESSEL FUNCTIONS
I:
FIGURE 9.8. e-zlo(z),e-zIl(~),eZKO(;C) and e"Kl(z).
FIGURE 9.9. 1,(5) and K,(5).
Relations Between Solutions
9.6.2 K( z )=h I-,(z) sin -I,(z) (y.)
The right of this equation is replaced by its limiting value if v is an integer or zero.
9.6.5 Y,(zeW) =et(,+l)riI ,( z 1 - (2/~)e-+*~K,(z)
(-*<a% z<h)
9.6.6 I-,(z)=l,(~), K-,(z)=K,(z)
Most of the properties of modified Beasel functions can be deduced immediately from those of ordinary Bessel functions by application of these relations.
Limiting Forms for Small Arguments
When v is fked and z+O
9.6.7
Iv(+(iz)yr(v+i) (vz -1, -2, . . .)
9.6.8 Ko(z)--ln z
9.6.9 K,(z)-+r (V)(~Z)- ' ( g v>o )
Ascending Series
o (42")" 1,(2)=(42)v 2 myv+k+i)
Kn(z>=&(34-" go k! (-322))"
9.6.10
9.6.11 n-1 (n-k-l)!
+ (-In+1 In (34In(~) (tz")" +(->"3(3d" 2 INC+l)+W+k+l) 1 k!(n+k)!
%”, denotes I”, eurfKv or any linear combination of these functions, the coefficients in which are independent of z and v.
I;(Z) = rl (z), K; (2) = - K~ (z)
Formulas for Derivatives
Analytic Continuation
9.6.30 Iu(zemrf) =em”’‘Iu(z) (m an integer)
9.6.31
Kv( t) = e-mmf Kv(z)--?ri sin (mvn) csc (v?r)I,(z) (m an integer)
9.6.32 I.(Z)=I.(z), K,.(B)=K,(z) (V real) - -
Generating Function and Associated Series
9.6.33 2 tkIk(z) (t#O) k=-m
m
9.6.34 ez cOse=Io(z) +2 C Ik(z) cos(k0) k-1
+2 2 k=l ( - ) ~ ~ ~ ( z ) cOs(2ke)
9.6.36 l=Io(~)-212(~)+214(~)-21~(~)+ . . .
9.6.37 ez=Io(z)+211(2)+212(2)+213(2)+ . . ,
9.6.38 e-z=Io(z)-~11(2)+212(2)-~13(2)+ . .
9.6.39 cosh ~=I~(z)+21~(~)+21,(~) +216(2)+ . . .
9.6.40 sinh 2=211(2)+213(z)+21~(2)+ . . . *See page 11.
BESSEL FUNCTIONS OF INTEGER ORDER 377 Other Werential Equations
The quantity X2 in equations 9.1.49 to 9.1.54 and 9.1.56 can be replaced by -A2 if at the same time the symbol W in the given solutions is replaced by 3.
9.6.41 zzw" + z( 1 f 2 2) w ' + ( f 2- S)w=O, w =e~2f2",( z)
Differential equations for products may be obtained from 9.1.57 to 9.1.59 by replacing z by iZ.
Derivatives With Reepect to Order 9.6.42
9.6.43
9.6.46
9.6.46
Expreesions in Terms of Hypergeometric Functions
9.6.47
9.6.48
OF^ is the generalized hypergeometric function. For M(a, b, z), Mo,,(z) and Wo,,(z) see chapter 13.)
Connection With Legendre Functions
If LL and z are fixed, Wz>O, and v+w through real positive values
9-6-49 lim{vre
9.6.50 E m { v-pe-p"
For the definition of P;" and Qf, see chapter 8.
Multiplication Theorems 9.6.51
Zeros
Properties of the zeros of I,(z) and K,(z) may be deduced from those of J,(z) and Hf)(z) respec- tively, by application of the transformations 9.6.3 and 9.6.4.
For example, if v is real the zeros of IJz) are all complex unlese -2k<v<- (2k- 1) for some posi- tive integer k, in which event I,(z) has two real zeros.
The approximate distribution of the zeros of K,,(z) in the region -#r<arg z s a r i s obtainedon rotating Figure 9.6 through an angle -3r so that the cut lies along the poaitive imaginary axis. The zeros in the region -$a <arg z 53% are their conjugates. K,,(z) has no zeros in the region larg zI <$a; this result remains true when n is replaced by any real number v.
The general terms in the last two expansions can be written down by inspection of 9.2.15 and 9.2.16.
If Y is real and non-negative and z is positive the remainder after k terms in the expansion 9.7.2 does not exceed the (k+l)th term in absolute value and is of the same sign, provided that k l v - - 3 .
9.7.5
( lag Z K b ) The general terms can be written down by inspection of 9.2.28 and 9.2.30.
Uniform A~wptot ie Expandons for Large Orders
9.7.8
9.7.10
When v++ m , these expansions hold uniformly with respect to z in the sector larg z1S&r--t, where e is an arbitrary positive number. Here
and Uk(t), q(t) are given by 9.3.9, 9.3.10, 9.3.13 and 9.3.14. See [9.38] for tables of q, uk(t), vr(t), and also for bounds on the remainder terms in 9.7.7 to 9.7.10.
Relations Between Solutions 9.9.5 ber-, z=cos(vr) ber, z+sin(vx) bei, z
+ (2/r) sin(ur) ker, z
bei-, z= -sin(~r) ber, z+cos(vr) bei, z + (2/r) sin(vr) kei, z
9.9.6 ker-. X=COS(VT) ker, z-sin(vx) kei, r ke i , z=sin(vx) ker, z+cos(vr) kei, z
9.9.7 ber-, a= (-)" ber, 1, bei-, x= (-)" bei, z
9.9.8 ker-, x= (-)" ker, r, kei-, z= (-)" kei, z
Ascending Seriee
9.9.9
9.9.10 (4g)2 I (4.")' . . . ber z=l-- (2!)2 (4!)2
380 BESSEL FUNCTIONS OF INTEGER ORDER
(n-k- l)! k! (t$)k-ln (3x) bei, x-4.. ber, x
where #(n) is given by 6.3.2.
9.9.12 ker x=-ln (42) ber x+tx bei x
kei x= --In ($2) bei x-tr ber x
Functions of Negative Argument
In general Kelvin functions have a branch point at z=O and individual functions with argu- ments zefri are complex. The branch point is absent however in the case of ber, and bei, when v is an integer, and
9.9.13
ber,(-z) = (-)” ber, z, bei,(-x) = (-)* bei, z
Recurrenee Relations
9.9.14
v 1 j :- ;jv=- Vv+l+gu+l)
f:+,f,=--- Cf,-,+g,-,>
Jz V 1
Jz where
9.9.15 f,=ber, XI j,=bei, x ‘1 gv=bei, x J j,=ker, x
g,=kei, x
g,=-ber, x J f,=kei, x
g,=-ker, x
9.9.16 ber‘ x=ber, x+beil z
bei’ x=-ber, x+bei, x 9.9.17
8 ker’ x=kerl x+kei, x
8 kei’ x=-kerl x+kei, x
Recurrence Relations for Goas-Products If
9.9.18 p.=bee x+beil x q,=ber, x bei: x-ber: x bei. z r,=ber, x ber: z+bei, x bei: x s.=ber? x+bei? x
then
9.9.19
Y
~;+1=--2p,+r,=--q,-,+2r,
and
9.9.20 pP&=8+d,
The same relations hold with ber, bei replaced throughout by ker, kei, respectively.
Indehite Integrals
In the following j,, g, are any one of the pairs given by equations 9.9.15 and jf, gf are either the same pair or any other pair.
9.9.21
9.9.22
BESSEL FTJNCTIONS OF INTEGER ORDER 381
5. 84892 7. 23883 11. 67396 16. 11356 20. 55463
9.9.24 1 J- x (f”g: +gy3CZ)dx=4 x2 (2fYg: - f v - 19*s+1
J- x (3-r g3 dx= z(fvg:-flg”)
-fy+1g2-1+2g.f*s-gv-,f~+1-g.+1f*s-1) 9.9.25
= - < z / a (f If ”+ 1 + g&+ 1 -fYgr+ 1. +f Y + 1 gv)
9.9.26 1 J zf ,gPdx=~ X V f 4 p - f P.-lg”+l-j”+lgr-l)
J X V : - sZ)dx=z 9 (f”y-fv- 1f.+1- gf + Y Y - 19.+1)
9.9.27 1
Ascending Series for Cross-Producta 9.9.28 be s x+bei: x=
5. 02622 9. 45541 13. 89349 18.33398 22. 77544
9.9.29
ber, x bei: x-ber: x bei, x
6. 03871 10. 51364 14. 96844
9.9.30
ber, x ber: x+bei, x bei: x
3. 77320 8. 28099 12. 74215
9.9.31
bedz x+beiL2 x
Expansions in Series of Bessel Functions
9.9.32
V + d 4 m e ( a ~ + k ) r i 1 4 ~ J
2tk k! ber, x+i bei, x = C k =O
9.9.33
Zeros of Functions of Order Zero 6 ~ I berz I bei x I kerz I keix
1st zero 2nd zero 3rd zero 4th zero 5th zero
1st zero 2nd zero 3rd zero 4th zero 5th zero
ber’x I bei‘x
19. 41758 17. 19343 23. 86430 21. 64114
1. 71854 6. 12728 10. 56294 15. 00269 19. 44381
ker‘ x
2. 66584 7. 17212 11. 63218 16. 08312 20. 53068
-
3.91467 8. 34422 12. 78256 17. 22314 21. 66464
kei’ z
4.93181 9. 40405 13.85827 18. 30717 22.75379
9.10. Asymptotic Expansions Asymptotic Expansions for Large Arguments
When Y is fixed and x is large
9.10.1
ber, x=- {f,(x) cos a+gv(z) sina} &d2
42Tx 1 -- {sin ( 2 4 ker, x+cos (2vn) kei, x} n-
9.10.2
bei, x = r t f , ( x ) sin a-g,(x) cos a} e‘lJ2 ,
*X
1
n- +- {cos ( 2 4 ker, x-sin ( 2 4 kei, x}
9.10.3 ker, x=dme - z l d 2 { f v ( - x ) cos B-g,(-x) sin
6 From British Association for the Advancement of Science, Annual Report (J. R. Airey), 254 (1927) with permissioL. This reference also gives 5-decimal values of the next five zeros of each function.
382 BESSEL FUNCTIONS
9.10.7
df 4 sin ($) ( - I ) (p -g) . . .{p-(2k-I)*}
k! (82) k
The terms" in ker,x and kei,x in equations 9.10.1 and 9.10.2 are asymptotically negligible compared with the other terms, but their inclusion in numeri- cal calculations yields improved accuracy.
The corresponding series for ber; x, b ei: x, ker: x and kei: x can be derived from 9.2.11 and 9.2.13 with ~ = x e ~ ~ ' l ~ ; the extra terms in the expansions of ber: x and bei; x are respectively
-(1/~) {sin(2vlr)ker; s+cos(2vr)kei: 2)
( I/T) {cos ( 2 ~ 4 ker: s-sin (2vlr) kei: 2) .
Modulus and Phase
-2 e)k Cc k-1
and
9.10.8 M,=,/(be~z+bei:x), @,=arctan (bei,x/ber,r)
9.10.9 ber,x=M, cos e,, bei,x=M, sin 8,
9.10.10 M-,=M,, e-,=e,--nT
9.10.11 ber: jC=#M,+, cos (O,+l-i+$ M,-l cos (L1-&r)
= (y/~)M. COS e,+M,, COS (eV+,-id = - (v/~)M, COS e,-M,-, COS (e,-,-+T)
9.10.12 bei: x= #M,+, sin (e,+, - 4.) - #M,-l sin (e,-l - tr)
=(~/x)M,sin e,+&, sin (e,,,-4T) = - (v/$)M,sin e,-M,-, sin (e,-,-+r)
9.10.13 ber' z=Ml cos (O1-fr),
9.10.14
bei' x=Ml sin (el-i?r)
M;= (V/Z)M,+M,+, COS (e,,-e,-ir> = - ( v / ~ ) ~ , - ~ , - l COS (e,-l-e,-+T)
e:= (M,+JM,) sin (e,+,-e.-44 9.10.15
= - (M,-l/M,) sin (e,-l-e,-tT)
8 The coefficients of these terms given in [9.17] are in- correct. The present results are due to Mr. G. F. Miller.
OF INTEGER ORDER
9.10.16 M;=Ml COS (el-eo-tlr) e;=(Ml/Mo) sin (el-eo-+r)
9.10.17
d(rwe;)/d;c=2w, ~ ~ ; ' + r ~ ; - - ~ , = ~ ~ d ; ~
9.10.18
N,=d(kee3 +kei:x), $,=arctan (kei,z/ker,x)
9.10.19 ker,s=N, cos $,, kei,z=N, sin 4,
FIGURE 9 . 1 1 . In Mo(x), eo(x), I n No(x) an& 40(x).
Equations 9.10.11 to 9.10.17 hold with the symbols b, M, e replaced throughout by k, N, 4, respectively. In place of 9.10.10
9.10.20 N_,=N,, &,=$.+vr
BESSEL FDNCTIONS OF INTEGER ORDER 383 Asymptotic Expansions of Modulus and P h e
kei’ x= -ln (32) bei’ 2 - P bei x - f ~ ber’ x - 4-4.21 139 217 - 13.39858 846(~/8)~ + 19.41 182 758(~/8)”-4.65950 823(~/8)’* f.33049 424[~/8)’”- .00926 7 0 7 ( ~ / 8 ) ~
+ .OOO 1 1 997 (x/8) +e
le1 <7 x 10-8
9.11.9 85x<-
ker x+i kei x=j(x)(l+tl)
9.11.10 81x< w
i her x+i bei x-- (ker x+i kei x)=g(x)(1+s2) T
lC1<3X 10-7
Numerical
I 9.12. Use and Extension of the Tablee
Examplel. To evaluate Jn(1.55), n=O, 1, 2, . ., each to 5 decimals. The rdcurrence relation
Jm-1 (z) + Js+l(~) = (2n/x)Js(x) I can be u s 4 to compute Jo(x), J1(x), J&), . . ., successively provided that n<x, otherwise severe accumulation of rounding errors will occur. Since, however, Jn@) is a decreasing function of n when n>x, recurrence can always be carried out in the direction of decreasing n.
Inspection of Table 9.2 shows that Jn(1.55) vanishes to 5 decimals when n>7. Taking arbi- trary values zero for Jo and unity for J8, we compu te by recurrence the entries in the second column of the following table, rounding off to the nearest integer at each step.
We normalize the results by use of the equation 9.1.46, namely
J~(x)+~J~(x)+~J~(x)+ . . . =I
This yields the normalization factor
1/322376=.00000 31019 7
BESSEL FUNCTIONS OF INTEGER ORDER 386 and multiplying the trial values by this factor we obtain the required results, given in the third column. As a check we may verify the value of Jo( 1.55) by interpolation in Table 9.1.
(i) In this example it was possible to estimate immediately the value of n=N, say, at which to begin the recurrence. This may not always be the case and an arbitrary value of Nmay have to be taken. The number of correct signifi- cant figures in the final values is the same as the number of digits in the respective trial values. If the chosen N is too small the trial values will have too few digits and insufficient accuracy is obtained in the results. The calculation must then be repeated taking a higher value. On the other hand if N were too large unnecessary effort would be expended. This could be offset to some extent by discarding significant figures in the trial values which are in excess of the number of decimals required in J,,.
(ii) If we had required, say, J0(1.55), J1(1.55), . . ., J10(1.55), each to 5 significant figures, we woiild have found the values of J10(1.55) and Jl1(1.55) to 5 significant figures by interpolation in Table9.3 and then computed by recurrence J8, Js, . . ., Jo, no normalization being required.
Alternatively, we could begin the recurrence at a higher value of N and retain only 5 significant figures in the trial values for n110.
(iii) Exactly similar methods can be used to compute the modified Bessel function I,,@) by means of the relations 9.6.26 and 9.6.36. If x is large, however, considerable cancellation will take place in using the latter equation, and it is preferable to normalize by means of 9.6.37.
Example 2. To evaluate Y,,(1.55), n=O, 1, 2, . . ., fO, each to 5 significant figures.
The recurrence relation
Remarks.
Y,,-l(4 + Y,,+l(4 = (244 Y,,(Z) can be used to compute Y,,(z) in the direction of increasing n both for n<x and n>x, because in the latter event Y,,(x) is a numerically increasing function of n.
We therefore compute Yo( 1.55) and Yl( 1.55) by interpolation in Table 9.1, generate Yz(l .55), Ya(1.55)) . . ., Ylo(1.55) by recurrence and check Ylo(1.55) by interpolation in Table 9.3.
n Y,(1.66) n Y,(1.66) 0 +O. 40225 6 -1.9917X10' 1 -0.37970 7 -1.5100XlW 2 -0.89218 8 -1.3440XlO' 3 -1.9227 9 -1.3722XlW 4 -6.5505 10 - 1.6801 X 10'' 6 -31.886
Remarks. (i) An alternative way of computing Yo(z), should Jo(x), Jz(x), J4(z), . . ., be avail- able (see Example 1)) is to use formula 9.1.89. The other starting value for the recurrence, Yl(x), can then be found from the Wronskian relation Jl(x) Yo(x) -Jo(z) Yl(x) =2/(7rz). This is a convenient procedure for use with an automatic computer.
(ii) Similar methods can be used to compute the modified Bessel function K,,(z) by means of the recurrence relation 9.6.26 and the relation 9.6.54, except that if x is large severe cancellation will occur in the use of 9.6.54 and other methods for evaluating Ko(x) may be preferable, for example, use of the asymptotic expansion 9.7.2 or the poly- nomial approximation 9.8.6.
Example 3. To evaluate J0(.36) and Y0(.36) each to 5 decimals, using the multiplication theorem.
From 9.1.74 we have
We take 2=.4. Then h=.9, (A2-1)(~z)=-.038, and extracting the necessary values of Jk(.4) and Yk(.4) from Tables 9.1 and 9.2, we compute the required results as follows: k a k akJk(.I) akYk(*4 ) 0 +1.0 + .96040 - .60602 1 4-0.038 + .00745 - .06767 2 +0.7220X lo-' + .oooo1 - .00599 3 +0.914Xl0-6 - .00074 4 +0.87x10-7 - .o0011 5 +0.7X10-0 - .woo2
Jo(.36) + .96786 Y0(.36) = - .68065
Remark. This procedure is equivalent to inter- polating by means of the Taylor series
at z=.4, and expressing the derivatives VAk)(z) in terms of g k ( z ) by means of the recurrence rela- tions and differential equation for the Bessel functions.
Example 4. To evaluate Jv(z), J;(x), Y,(x) and X(x) for v=50, 2=75, each to 6 decimals.
We use the asymptotic expansions 9.3.35, 9.3.36, 9.3.43, and 9.3.44. Here z=x/v=3/2. From 9.3.39 we find
1 2 2 (-c)a'z=Z &-arccos -=+.2769653. 3 3
BESSEL FUNCTIONS OF INTEGER ORDER 387 Hence
{=-.5567724 and ( 2y4=+1 . 155332 . 1-22
Next,
Interpolating in Table 10.11, we find that
v113=3 .68403l, v213{= -7.556562.
Ai(v213{)= + .299953,
Bi(v2I3{) = - .160565,
Ai‘(v213{) = + .451441,
Bi’(v2I3{) = + 319542.
As a check on the interpolation, we may verify that Aj Bi’- Ai‘Bi = 1 IT.
Interpolating in the table following 9.3.46 we obtain
bo( {) = + -0136, co({) = + .1442.
The contributions of the terms involving al({) and dl({) are negligible, and substituting in the asymptotic expansions we find that
Remarks. This example may also be computed using the Debye expansions 9.3.15,9.3.16, 9.3.19, and 9.3.20. Four terms of each of these series are required, compared with two in the computations above. The closer the argumentader ratio is to .unity, the less effective the Debye expansions become. In the neighborhood of unity the expan- sions 9.3.23, 9.3.24, 9.3.27, and 9.3.28 will furnish results of moderate accuracy; for high-accuracy work the uniform expansions should again be used.
Example 5. To evaluate the 5th positive zero of Jlo(z) and the corresponding value of Jio(z), each to 5 decimals.
We use the asymptotic expansions 9.5.22 and 9.5.23 setting v=10, s=5. From Table 10.11
we find
aa= -7.944134, &‘(a,) = + .947336.
Hence
Interpolating in the table following 9.5.26 we obtain
The bounds given at the foot of the table show that the contributions of higher terms to the asymptotic series are negligible. Hence
jlo,6=28.88631+.00107+ . . . =28.88738,
2 .947336 J’0ci10*6)=-102/5 2.888631 X .98259
X(l-.00001+ . . .)=-.14381.
Example6. To evaluate the first root of Jo(z)Yo(XZ)-Yo(z)Jo(XZ)=O for A=# to 4 signifi- cant figures.
Let a:’) denote the root. Direct interpolation in Table 9.7 is impracticable owing to the divergence of the differences. Inspection of 9.5.28 suggests that a smoother function is (A-1)ai1). Using Table 9.7 we compute the fol- lowing values
1/x (A- l,ap) 6 0 0.4 3.110
0.6 3.131
0. 8 3.140
1.0 3.142(~)
-12
-7
+21
+9
$2
Interpolating for l/A= .667, we obtain (A-l)ai1)=3.134 and thence the required root ai.’! = 6.268.
Example 7. To evaluate ber, 1.55, bei, 1.55, n=O, 1, 2, . . ., each to 5 decimals.
We use the recurrence relation
Jn-1(2e3r114) +J n+l (ze3rf’4 1
taking arbitrary values zero for J9(xe3rf/4) and l + O i for J8(zP14) (see Example 1).
The values of ber,z and bei,,z are computed by multiplication of the trial values by the normal- izing factor
1/(294989 - 2201 li) = (.337119 + .025155i) X
obtained from the relation
Adequate checks are furnished by interpolating in Table 9.12 for ber 1.55 and bei 1.55, and the use of a simple sum check on the normalization.
Should ker,,z and kei,z be required they can be computed by forward recurrence using formulas 9.9.14, taking the required starting values for n=O and 1 from Table 9.12 (bee Example 2). If an independent check on the recurrence is required the asymptotic expansion 9.10.38 can be used.
References
Texte
[9.1] E. E. Allen, Analytical approximations, Math. Tables Aids Comp. 8, 240-241 (1954).
[9.2] E. E. Allen, Polynomial approximations to some modified Bessel functions, Math. Tables Aids Comp. 10, 162-164 (1956).
[9.3] H. Bateman and R. C. Archibald, A guide to tables of Bessel functions, Math. Tables Aids Comp. 1, 205-308 (1944).
[9.4] W. G. Bickley, Bessel functions and formulae (Cambridge Univ. Press, Cambridge, England, 1953). This is a straight reprint of part of the preliminaries to [9.21].
[9.5] H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford Univ. Press, London, England, 1947).
[9.6] E. T. Copson, An introduction to the theory of functions of a complex variable (Oxford Univ. Press, London, England, 1935).
Higher transcendental functions, vol 2, ch. 7 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953).
[9.8] E. T. Goodwin, Recurrence relations for cross- products of Beasel functions, Quart. J. Mech. Appl. Math. 2, 72-74 (1949).
[9.9] A. Gray, G. B. Mathews and T. M. MacRobert, A treatise on the theory of Bessel functions, 2d ed. (Macmillan and Co., Ltd., London, England; 1931).
[9.10] W. Magnus and F. Oberhettinger, Formeln und Satze fiir die speziellen Funktionen der mathe- matischen Physik, 2d ed. (Springer-Verlag; Berlin, Germany, 1948).
[9.11] N. W. McLachlan, Bessel functions for engineers, 2d ed. (Clarendon Press, Oxford, England, 1955).
[9.12] F. W. J. Olver, Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. SOC. 48, 414-427 (1952).
[9.7] A. Erd6lyi et al.,
[9.13] F. W. J. Olver, The asymptotic expansion of Bessel functions of large order. Philos. Trans. Roy. SOC. London A241, 328-368 (1954).
[9.14] G. Petiau, La th6orie des fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, France, 1955).
[9.15] G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, Cambridge, England, 1958).
[9.16] R. Weyrich, Die Zylinderfunktionen und ihre Anwendungen (B. G. Teubner, Leipzig, Germany, 1937).
[9.17] C. 5. Whitehead, On a generalization of the func- tions ber z, bei 2, ker 2, kei 2. Quart. J. Pure Appl. Math. 42, 316-342 (1911).
[9.18] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952).
Tables
(9.191 J. F. Bridge and S. W. Angrist, An extended table of roots of JA(z) Yi(@z) -J:(@z) Yi(z)=O, Math. Comp. 16, 198-204 (1962).
[9.20] British Association for the Advancement of Science, Bessel functions, Part I. Functions of orders zero and unity, Mathematical Tables, vol. VI (Cambridge Univ. Press, Cambridge, England, 1950).
[9.21] British Association for the Advancement of Science, Bessel functions, Part 11. Functions of positive integer order, Mathematical Tables, vol. x (Cambridge Univ. Press, Cambridge, England, 1952).
[9.22] British Association for the Advancement of Science, Annual Report (J. R. Airey), 254 (1927).
[9.23] E. Cambi, Eleven- and fifteen-place tables of Bessel functions of the first kind, to all significant orders (Dover Publications, Inc., New York, N.Y., 1948).
BESSEL FUNCTIONS OF INTEGER ORDER 389
[9.24] E. A. Chistova, Tablitsy funktsii Besselya ot deistvitel’nogo argumenta i integralov ot nikh (Izdat. Akad. Nauk SSSR, Moscow, U.S.S.R., 1958). (Table of Bessel functions with real argument and their integrals).
[9.25] H. B. Dwight, Tables of integrals and other mathe- matical data (The Macmillan Co., New York, N.Y., 1957).
This includes formulas for, and tables of Kelvin functions.
[9.26] H. B. Dwight, Table of roots for natural frequencies in coaxial type cavities, J. Math. Phys. 27,
This gives zeros of the functions 9.6.27 and 9.6.30
[9.27] V. N. Faddeeva and M. K. Gavurin, Tablitsy funktsii Besselia J.(z) tselykh nomerov ot 0 do 120 (lzdat. Akad. Nauk SSSR, Moscow, U.S.S.R., 1950). (Table of J,(z) for orders 0 to 120).
[9.28] L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal Society Shorter Mathematical Tables No. 3 (Cambridge Univ. Press, Cambridge, England, 1954).
[9.29] E. T. Goodwin and J. Staton, Table of J3G0..t), Quart. J. Mech. Appl. Math. 1, 220-224 (1948).
[9.30] Harvard Computation Laboratory, Tables of the Beasel functions of the first kind of orders 0 through 135, vola. 3-14 (Harvard Univ. Press, Cambridge, Mass., 1947-1951).
[9.31] K. Hayashi, Tafeln der Besselschen, Theta, Kugel- und anderer Funktionen (Springer, Berlin, Ger- many, 1930).
[9.32] E. Jahnke, F. Emde, and F. Loach, Tablea of higher functions, ch. IX, 6th ed. (McGraw-Hill Book Co., Inc., New York, N.Y., 1960).
[9.33] L. N. Karmazina and E. A. Chistova, Tablitsy funktsii Besselya ot mnimogo arguments i integralov ot nikh (Izdat. Akad. Nauk SSSR, Moscow, U.S.S.R., 1958). (Tables of Bessel
84-89 (1948).
for n=0,1,2,3.
functions with imaginary argument and their integrals).
[9.34] Mathematical Tables Project, Table of fn(z)=nl(Hz)-nJ.(z). J. Math. Phys. 23, 45-60 (1944).
[9.35] National Bureau of Standards, Table of the Bessel functions Jo(z) and JI(z) for complex arguments, 2d ed. (Columbia Univ. Press, New York, N.Y., 1947).
[9.36] National Bureau of Standards, Tables of the Bessel functions Yo@) and Yl(z) for complex arguments (Columbia Univ. Press, New York, N.Y., 1950).
[9.37] National Physical Laboratory Mathematical Tables, vol. 5, Chebyshev series for mathematical func- tions, by C. W. Clenshaw (Her Majesty’s Sta- tionery Office, London, England, 1962).
[9.38] National Physical Laboratory Mathematical Tables, vol. 6, Tables for Bessel functions of moderate or large orders, by F. W. J. Olver (Her Majesty’s Stationery Office, London, England, 1962).
[9.39] L. N. Nosova, Tables of Thomson (Kelvin) functions and their first derivatives, translated from the Russian by P. Basu (Pergamon Press, New York, N.Y., 1961).
[9.40] Royal Society Mathematical Tables, vol. 7, Bessel functions, Part 111. Zeros and associated values, edited by F. W. J. Olver (Cambridge Univ. Press, Cambridge, England, 1960).
The introduction includes many formulas con- nected with zeros.
[9.41] Royal Society Mathematical Tables, vol. 10, Bessel functions, Part IV. Kelvin functions, by A. Young and A. Kirk (Cambridge Univ. Press, Cambridge, England, 1963).
The introduction includes many formulas for Kelvin functions.
19.421 W. Sibagaki, 0.01 ’% tables of modified Bessel functions, with the account of the methods used in the calculation (Baifukan, Tokyo, Japan, 1955).
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