2. 356 BESSEL FTJNCTIONS OF INTEGER ORDER
Page
Table 9.3. Bessel Functions-Orders 10, 11,20, and 21 ( 0 5 ~ 1 2 0 ) .
. 402
2-*0J1o(x), z-l'J11 (z), z'"1o(z)
z=O(.l)lO, 8 or 9 s s
JlO(Z>, J l l ( 4 , YlO(Z>
2=10(.1)20, 8D
2-'0J20(2), 22 J 1 z , 2 V 2 0 ( 2 )
- 1 2 ()
z=0(.1)20, 6 or 7
s s
Bessel Functions-Modulus and Phase of Orders 10,11,20,
and 21 ( 2 0 5 ~ Q)). . . . . . . . . . . . . . . . .
5 406
zQ&dz), e n ( 4 -2
n=10, 11, 8D
n=20, 21, 6D
z-'= .05(- .002)0
Table 9.4. Bessel Functions-Various Orders (OIn1100). . . . . . 407
Jn(z),Y&), n=0(1)20(10)50, 100
z=1, 2, 5, 10, 50, 100, 10s
Table 9.5. Zeros and Associated Values of Bessel Functions and Their
Derivatives ( O s n 1 8 , 1 5 ~ 1 2 0 .) . . . . . . . . . . 409
j w , JX.in.8) ; j;*s, Jn(j;,8), 5D (10D for n=o>
ynS8, y n J ; Y ; . ~ , YJY;,~), 5D (8D for n=o>
W
s=1(1)20, n=0(1)8
Table 96
.. Bessel Functions Jo(jo.,z), . . . . . . . . . . .
s= 1 (1)5 413
z=O(.O2)1, 5D
Table 9.7. Bessel Functions-Miscellaneous Zeros (s=1(1)5) . . . . . 414
8th zero of z J1 -Wdz)
(2)
X, X-'=O(.O2) .I, .2(.2)1, 4D
8th zero of Jl(z) -Wo(z)
X= .5(.1) 1, X-'= 1 (- .2).2, .I (- .O2)0, 4D
8th zero of Jo(z) Yo(Az)Yo(z)Jo(Az)
-
X-'=.8(-.2) .2, .1(-.02)0, 5D (8D for s=1)
8th zero of Jl(z) Yl(Az)- Y,(z)J1(Az)
X-l=.8(--.2) .2, .1(-.02)0, 5D (8D for s=1)
8th zero of J1(z) Yo(Az)- Yl(z)Jo(Az)
X-'=.8(--.2) .2, .1(-.02)0, 5D (8D for s=1)
Table 9.8. Modified Bessel Functions of Orders 0, 1, and 2 (0 1 2 1 2 0 ) . 416
e-zlo(z), ezKo(z),e+ll( z), ezKl(z)
z=O(.l)lO (.2)20, 10D or 10s
Z-~Z(~),
2Kab)
~=0(.1)5, 10D, 9D
e-zla (2), e'& (5)
z=5(.1)10 (.2)20, 9D, 8D
Modified Bessel Functions-Auxiliary Table for Large
Arguments ( 2 0 5 ~ Q)) . . . . . . . . . .
5 . . . . . 422
de-zln(z), n-ldeZK.(z), n=O, 1, 2
z-'= .05(-.002)0, 8-9D
Modified Bessel Functions-Auxiliary Table for Small
Arguments ( 0 5 ~ 1 2 ) .. . . . . . . . . . . . . . . 422
Ko(z)+Io(4 lnz, 4&(4--11(4 hzl
z=0(.1)2, 8D
3. BESSEL FUNCTIONS OF INTEGER ORDER 357
Page
Table 9.9. Modified Bessel Functions-Orders 3-9 (O<z<20) . . . 423
e-”1,(z), e”K,(z), n= 3 (1)9
Z= O( .2)10(.5)20, 5s
Table 9.10. Modified Bessel Functions-Orders 10, 11, 20 and 21
( < <0
OS2 ) . . . . . . . . . . . . . . . . . . . . 425
z-11111(2),
z-’Ol10(z), zloKlo(z)
z=0(.2)10, 8 or 9 s
s
e-zllo(z), e-”111(z), e”Kd4
z=10(.2)20, 10D, lOD, 7D
z-zo120(z),z-z1121(4, Z0Kz0(z)
z=0(.2)20, 5 s to 7 s
Modified Bessel Functions-Auxiliary Table for Large
Arguments (20 <z < a). . . . . . . . . . . . . . . 427
In{z+e-zllo(z)}, ln{z+e-zIll(z)},ln{?r-lzfeZKlo(z)}
In{z+e-zlzo(z)}, z+e-zlzl(z)},
In{ ln{r-lz’eZKzo(z)}
~-‘=.05(-.001)0, SD, 6D
Table 9.11. Modified Bessel Functions-Various Orders (0 <n <100) . 428
In(z),K,,(z), n=0(1)20(10)50, 100
z=1, 2, 5, 10, 50, 100, 9s or 10s
Table 9.12. Kelvin Functions-Orders 0 and 1 ( << )
Oz5 . . . . . . 430
ber z, bei z, berl 2, bei, z
ker z, kei z, kerl z, keil z
z=0(.1)5, IOD, 9D
Kelvin Functions-Auxiliary Table for Small Arguments
(O<z<l). . . . . . . . . . . . . . . . . . . . . 430
ker z+ber z In z, kei z+bei z z I n
z(kerlz+berl z In z), z(kei, z+beil z z) I n
z=O(.1)1, 9D
Kelvin Functions-Modulus and Phase ( 0 1 ~ 1 7 ) . . .
. 432
Mo(4, eO(4, Ml(d, el(4
N O ( 4 , 40(4, N1(4,41(4
z=0(.2)7, 6D
Kelvin Functions-Modulus and Phase for Large Argu-
ments (6.6535 a ) . . . . . . . . . . . . . . . .
. 432
eo(z)
z+e-”’J2Mo(z),-(z/./z), z+e-Z/~Mlel(z)-(z/Jz)
(z),
z+ezlJ2No(z>, + (z/./z>,
40(4 zifez’.“N(z),(2)+ (z/./z)
$1
z-’= -.Ol)O, 5D
.15(
The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K.
Cherwinski, Elizabeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and
Ruth Zucker of the National Bureau of Standards, and N. F. Bird, C. W. Clenshaw, and
Joan M. Felton of the National Physical Laboratory in the preparation and checking of the
tables and graphs.
4. 9. Bessel Functions of Integer Order
Mathematical Properties
Notation Bessel Functions J and Y
The tables in this chapter are for Bessel func- 9.1. Definitions and Elementary Properties
tions of integer order; the text treats general
orders. The conventions used are: Differential Equation
z=z+iy; z, y real. d2w
22-+2 dw
-+(22-v2)w=O
n is a positive integer or zero. 9.1.1 dz2 dz
v, p are unrestricted except where otherwise
indicated; v is supposed real in the sections devoted Solutions are the Bessel functions of the f i s t kind
to Kelvin functions 9.9, 9.10, and 9.11. J*.(z), of the second kind Yv(z) (also called
The notation used for the Bessel functions is Weber’s function) and of the third kindH$”(z),H:z)(z)
that of Watson [9.15] and the British Association (also called the Hankel functions). Each is a
and Royal Society Mathematical Tables. The regular (holomorphic) function of z throughout
function Y,(z) is often denoted Nv(z) by physicists the z-plane cut along the negative real axis, and
and European workers. for fixed z ( f 0 ) each is an entire (integral) func-
Other notations are those of: tion of v. When v= &n, Jv(z)hrts no branch point
Aldis, Airey: and is an entire (integral) function of z.
Important features of the various solutions are
G,(z) for -+rY,(z),K,(z) for (-)*K,(z). as follows: Jv(z)(9?v20) is bounded as z+O in
any bounded range of arg z. Jv(z) and J-,(z)
Clifford:
are linearly independent except when v is an
C,(z) for ~-4~J,,(2fi). integer. J.(z) and Y ( ) linearly independent
, z are
for all values of v.
Gray, Mathews and MacRobert [9.9]: H!’)(z) tends to zero as IzI+- in the sector
Oag ;T<Z
<r Hi2)(z)tends to zero lzl--+m in the
as
Y,(z) for +rY,(z)+ On 2--r)J,(z), sector -r<arg O<.
z For all values of v, H!”(z)
- and H!”(z)are linearly independent.
Y,(z) for revr*sec(va) Y,(z),
0,(2) for +7riH:1)
(2).
Jahnke, Emde and Losch [9.32]: Relatione Between Solutions
a&) for r (Y+ 1)($2) - ’ (2).
J , J,(z) (m)- J-,(z)
COS
9.1.2 Y,(z)= sin (m)
Jeff reys:
The right of this equation is replaced by its
H.s,(z) for HY(z), Hi,(z) for H?)(z), limiting value if v is an integer or zero.
Kh,(z) for (2/a)K,(z). 9.1.3
Heine:
K (2) for- ~ P Y(2).
, ,
Neumann:
Yn(z) for )?rY,(z)+(ln 2--y)Jn(z).
Wbittaker and Watson [9.18]:
K,(z) for cos(vir)K,(z).
358
5. BESSEL FUNCTIONS OF INTEGER ORDER 359
L I
9.2. Ylo(x),
FIGURE Jlo(z), and
M o (z)=JJ:o (XI+ E (XI.
o
FIGURE Jo(z),YO@),
9.1. Jl(z), Yl(z>.
’ FIGURE J,.(lO) and y”(10)-
9.3.
FIGURE Contour lines of the modulus and phase of the Hankel Function HP(x+iy)=MoefSo. From
9.4.
E. Jahnke, F. Emde, and F. Losch, Tables of higher functions, McGraw-Hill Book CO.,Inc., New
York, N.Y., 1960 (with permission).
6. 360 BESSEL FUNCTIONS OF INTEGER ORDER
Limiting Forms for Small Arguments Integral Representations
9.1.18
When v is fixed and z+O
9.1.7
J~ =;
(2)
1
S,’ cos (z sin
(VZ-1, -2, -3, . . .) 9.1.19
Jv(z)-($z)v/r(V+l)
9.1.8 In z
Yo(z)--iH~1)(z)~~H~2)(z)~(2/~) ~,(z)=f I” cos (z cos e) {r+h sin2 e) 1 d~
(22
9.1.9 9.1.20
YJZ) - --i~:l) (2) - i ~ : 2 ) (z) -- ( I / ~r (.) ($2) -I
)
(9v>O)
Ascending Series
9.1.21
COS (zsin e-&)&
9.1.11
($E!)-” n-1 (n-k-l)!
Y,(z)=--
T k=O k! (tz”>”
9.1.22
where $(n)is given by 6.3.2.
9.1.13
9.1.26
In the last integral the path of integration must
lie to the left of the points t=O, 1, 2, . . . .
7. {
BESSEL FUNCTIONS OF INTEGER ORDER 361
and
4
9.1.34 pa,- qvr,=-gab
Analytic Continuation
In 9.1.35 to 9.1.38, m is an integer.
9.1.35 ze’” =em- J (
Jv( ,z)
9.1.36
=e-mvrfYv(z) sin(mvr) cot(vr) J,(z)
Y,(zemrf) +2i
9.1.37
sin (v~)H:~) = -sin
(amr f, sin(mvr) H!’)
(2)
}
(m- 1) v r H;l)(z)
9.1.38
(amr? (m+ 1) v r )2 : (z)
sin(vr)H;’) =sin 3’ )
+e v r f sin (mvr)H:’)
(z)
{{
9.1.39
-e-vrfH$’) 0
H!l)(zerf)=
H:’) = -p*Hil’(z)
(ze-rf)
9.1.40
- -
JIG)=Jv(z) Y,G) = Yv(z)
H;l)(Z)=m) Hr)(Z)=Hm (V real)
(k=O, 1,2, . . .)
9.1.31 Generating Function and A m i a t e d Serier
m
9.1.41 eW-W)=
k--m
t*Jk(Z) (tW
m
9.1.42 cos (z sin e)=Jo(z) +2 C Jzk(z) (2M)
cos
k=l
m
9.1.43 sin (z sin e)=2 Jnk+l(~)
sin (2k+l)O}
k-0
9.1.44
m
COS (Z COS e)=Jo(z)+2 (-)kJzk(~)
COS (2M)
k-1
9.1.45
m
sin (z cos e)=2 (-)kJ2k+l(z)cos (2k+l)B}
k-0
v v+l
P.+l+T,=; p,-- b P+
Vl
v v+l
rv+1+qv=- p,-- a P?+l
b
1 1 3
8.=2 P..+l+ZP.-I-& p, 9.1.48 sin z=2J~(z)-25,(2)+2J5(z)- . . .
8. 362 BESSEL FUNCTIONS OF INTEGER ORDER
Other Differential Equations
9.1.49 w" 4- ( x2--
3-
z2
3)w=o, w=zQ?,(hz)
Derivatives With Respect to Order
9.1.50 wf +(E--) 3-1 w=o,
9.1.a
{
9.1.51 W" +X22P-2~=0, W=Z+ %?llp(2X~fP/p)
9.1.52
9.1.65
2v-1
W''-- w1+X2w=O, w=z*%v(Az)
Z
9.1.53
b
21w" + (1 -2p)m' + (A7$9g+p=- 3$)w= 0, --c9c (4avJJZ)
- - r . (2)
7J
w =ZP%,(XZ~) (VZO, f l ,f2, . . , )
9.1.54 9.1.66
w" + (A2eZ2- v2)w= 0, w = U,(Ae2)
{
9.1.55
2(2-v2)w"+z(z~-33)w' 9.1.67
+ (22--3)2--(~+v~)}w=O, w=U:(z)
9.1.56
w(*n) (-)"A2nZ-nw,
= w= z*'Un(2Aazj) 9.1.68
where a is any of the 2n roots of unity.
DiEerentiaI Equations for Products
d Expreaaions in Terms of Hypergeometric Functions
In the following QE z -and U,(z), 9,(z) are any
dz
cylinder functions of orders v, p respectively. 9.1.69
9.1.57
94- w
2(v"+ s)tp2+ (9- p*yj
+422(9+ 1)(9+2)~=0, ?~=%',(z)52~(2)
9.1.70
9.1.58
Q(Q2-43)~+4zP(Q+1)~=0, ~=U.(z)gv(z)
as A, p+= through real or complex values; z, v
915
..9 being fixed.
Zaw"'+2(4~?+ 1-49)~'+(43- l)w=O, (oF1is the generalized hypergeometric function.
For M(a, b, z) and F(a, b;c; z) see chapters 13 and
w = zU,( 2) 9 I(2)
15.)
Upper Bounds
Connection With Legendre Functions
9.1.60 JJ. I I 1 (V>_O),
(2) 1 J, I I/&
(4 5 (v 2 1)
If p and z are fixed and v+- through real
positive values
9.1.71
1 =J,,(z) (z>O)
9. BESSEL F”CT1ONS OF INTEGER ORDER 363
9.1.72 In 9.1.79 and 9.1.80,
w=~(u2+2?-2uv cos a),
lim (I+”;’ (cos E)}=-4rY,(z) (~>0) u-v cos a=w cos x, v sina=w sin x
For P;’ and Q;”, see chapter 8. the branches being chosen so that 2o--vu and x+O
as v+O. C‘X)(cos is Gegenbauer’s
a) polynomial
Continued Fractions (see chapter 22).
9.1.73
J’(4
-
- 1 1 1 ...
J,-l(2)-2vz-’-- 2(v+1)2-’- 2(v+2)2-l--
-34.
-- iz2/{ b+1) (v+2) 1 , . .
y(v+1) 1 tz2/{
1- 1- 1-
Multiplication Theorem
9.1.74
G e g e n h w ’ saddition theorem.
WPr(Xz)=X*’ c m
k-0
g
(F)*(~*-1)*(3~)* (z)
k! ’33
If u, v are real and positive and 0 Sa Sa, then w, x
(IX’-ll<l) are real and non-negative, and the geometrical
relationship of the variables is shown in the dia-
If W= J and the upper signs are taken, the restric-
gram.
tion on X is unnecessary. Thc restrictions Ive*‘”l< 1 . 1 are unnecessary in
This theorem will furnish expansions of V W e ) 9.1.79 when g=J ,, is an integer or zero, and
and
in terms of Y,*k(r). in 9.1.80 when Y=J.
Addition Theorems Degenerate Form (u= 0):
Neumann’s
9.1.81
9.1.75 %‘,(uz!d= 5 %‘,~&)Jdv) (Ivl<lul)
et0 “06a=r(v)(32))-v 2 (v+k)i*J,+r(v)C:”(cosa)
k-0
ki-m ( Y Z O , -1, . . .)
The restriction l ll l
v<u is unnecessary when Neumann’sExpansion of an Arbitrary Function in e
%‘= and v is an integer or zero. Special cases are
J Series of Beasel Function8
9.1.76 l=JX2)+25 Z 2
( )
9.1.82 f(z)=aoJo(z)+2 2aJ&)
k-1
(Izl<c)
k-1
9.1.77 where c is the distance of the nearest singularity
off(z) from z=O,
o c
= 2n
(-)*Jk(Z)JP,-*(z> +2 5J*(z)Jz,+r(d (n21)
k-0 k-1
9.1.83 a*=-1
2a-i J +e’
f(t)O*(t)dt (CC
O’)
<<
9.1.78
and O,(t) is Neumann’spolynomial. The latter
c(-)*Jdz)J,+dZ)
m
Jn(2Z)=e Jr(Z)J,-i (2) 4-2 is defined by the generating function
k-0 k-1
Graf”e
9.1.84
9.1.79 L=JO(z)Odt)+2 k-1
t-2 5J&)odt> (Izl<tl)
%‘’(W)
cos
sin vX= k--m c W,+&)J*(v)
m
cos
s . ka(lveftal<lul) O,(t) isapolynomialof degreen+l in l/t; Oo(t)=l/t,
9.1.85
Gegenbauer’s
n(n-k-l)! 2 “-a+* (n=1,2,. . .)
9.1.80 w - 4 -’% ko
l k! (t>
-d ~ ) - ~ , ~ ( ~ (v+k)
*-
W’
) 2 W p + t ( ~ ) JP+$v)
U’ V
c(x’(cos The more general form of expansion
a)
k-0
(v#O,-l,. . ., Ive**al<IuI)
9.1.86 j(z) =ao~.(z>+2 5 aJv+*(z)
k-1
10. 364 BESSEL F"CT1ONS OF INTEGER ORDER
also called a Neumann expansion, is investigated 9.2.6
in [9.7] and [9.15] together with further generaliza- Yv(z)=~'2/(rz){P(v, sin x+Q(v, z) cosx}
z)
tions. Examples of Neumann expansions are
9.1.41 t 9.1.48 and the Addition Theorems. Other
o (la% 2 <
1
examples are 9.2.7
9.1.87 H,'"(z)=,/G){P(v,
z)+iQ(v, z)jefx
(-T<arg 2<2r)
(VZO, -1,-2,. . .) 9.2.8
9.1.88 (z)
H!2) = d m { P( z) -iQ(v, z) }e-fx
v,
n!($z)-" n-1
Y, (2) =--
(+Z>Vk(Z)
(-2a<arg z<r)
T (n-k)k! where x=z-(+++)u and, with 4v2denoted byp,
9.2.9
where +(n)is given by 6.3.2. + b-1) (p-9) (p-25) (p-49) - . . .
4! (82)'
9.1.89 9.2.10
-cc--l (~--~)(P-9~(cC--5)+ . . .
9.2. Asymptotic Expansions for Large 82 3! ( 8 ~ ) ~
Arguments If v is real and non-negative and z is positive, the
remainder after k terms in the expansion of P(v, z)
Principal Asymptotic Forms
does not exceed the (k+l)th term in absolute
When Y is fixed and lz1+co 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-
9.2.2 ceding subsection
Yl(4 =m 9.2.11
JL(z>=J~{ --~(v, z) sinx--S(v, z) cos x}
(la% zl<r)
9.2.12
Y:(z) =JG) x- S(v, z) sin x)
{ R(v, z) cos
(larg zl<r>
9.2.13
Hankel's Asymptotic Expansions
z)
H;l)'(z)= 42/(~z){iR(v, --S(v, z)}e'x
When v is k e d and 1zI+- (-7r<arg z2)
<r
9.2.5 9.2.14
J,(z)=J-a/(rz){P(v, cosx-QQ(v,2) sinx)
2) ~!2)'(z)=,/m{ - - i ~ ( v , z)--~(u, z))e-'X
I
( arg 4
I *<
(-2r<arg z<r)
11. BESSEL m C T I O N S OF INTEGER ORDER 365
9.2.15 9.2.29
{
(p- 1) (p-25) (p-1) (p2- 114p+1073)
+ 6 ( 4 ~ ) ~+ 5 (42)
9.2.16
1) (5p3- 1535p2+54703p-375733)
+ (p- 14(4.)’
+ . ..
9.2.30
X--{1--
2 --- 1 . 1 (c’-l)(Cc-45)-*.
1 p-3 *)
ax 2 (2x12 2 . 4 (2x14
Modulus and Phase
For real v and positive x The general term in the last expansion is given by
9.2.17 -1 - 1 3 . . . (2k-3)
M, = W (4I =.I{ c m x ) + E(2)
I 1 2 . 4 - 6 . . . (2k)
8,=arg H:’)(x)=arctan Y,(x)/Jv(x)) (p-l)(p-9). . .{/~-(2k-33)~}{p-(2k+1)(2k-l)~}
X
(22)
}
9.2.18 2k
N”=~H~”’(2)1=.I{JL’(x)+Y:’(x)} 9.2.31
(p, =arg Hi1) = arc tan Y (x)/ L
(x)’ L J(x)
p+3 p2+46p-63
9.2.19 J,(x)=M, cos e,, Y,(x)=M, sin e,, qJ”-X-(+v-;) f+
+
i-
2(4x) 6(4~)~
9.2.20 J:(x)=N, (pv,
cos Y:(x)=N,sin (pV. p3+ 185p2-2053p$ 1899
+ 5(42)
+.. .
In the following relations, primes denote differ-
entiations with respect to x. If v 20,the remainder after k terms in 9.2.28 does
9.2.21 M: =2/(a~)
:
e xv:(2-
=2 v2)/(if$) not exceed the (k+l)th term in absolute value
and is of the same sign, provided that k>v-$.
9.2.22 =M;2+A4;e:2 M:’
= +4/(rxMy)’
9.2.23 (2-v2)MPM~+ZN,N:XX
+ =O
9.3. Asymptotic Expansions for Large Orders
9.2.24
Principal Asymptotic Forms
tan 0 . =M,O:/M:=2/(axMyM~)
((py - )
M,N,sin ((pV-e,)=2/(ax) In the following equations it is supposed that
v+ OJ through real positive values, the other vari-
9-2-25 2M;’ xM:(2P)Mv-4/(11.2M:)
+ + - =O ables being fixed.
9.2.26
9.3.1
2w”’tx(42+ 1-4v2)w’+ (4v2- l)w=O, w=xM
Asymptotic Expansions of Modulus and Phase
When v is fixed, xis large and positive, and p=4v2 9.3.2
9.2.28 ,v(tsnh a-a)
12. 366 BESSEL FUNCTIONS OF INTEGER ORDER
9.3.3 9.3.9
J (v sec @) =
.
J2/(7rv tan 8) {cos (v tan p- vp- &)+ O (v-1)
<o<a<a*>
} uo(t)=l
~l(t)=(3t-5t?/24
UZ (t)== (8lt2-462t4+ 385t6)/l152
u3(t)=(3O375t3-3 69603t5+7 65765t7
Y.(v sec p)= -4 25425t9)/4 14720
~ 4 ( t = (44 651255'- 941 21676ts+3499 22430t'
)
J2/(?rv tan @) {sin (v tan p-vp-$n)+O(v-*)}
-4461 85740t"+ 1859 10725t12)/398 13120
(O<P<h)
9.3.4 For u5(t) and u,(t) see [9.4] or [9.21].
J& +ZV%)=2%-46 Ai( -2542)+O(v-l) 9.3.10
Y,(v+zvB)=-2!%-H Bi(-Pz) +O(v-l) Uk+l(t) = it'( 1- tz)>d (t)+g1 ' (1-5t2)u,(f)dt
0
(k=O, 1 , . . .)
9.3.5 Also
2% 1 9.3.11
Y&)--- -
3 W ($) d 4
J:(v sech a) -
{
9.3.12
Y:(v sech a)
k=l
1
where
9i3.13
vo(t)=l
v,(t)= (-9t+7f!)/24
+
vz(t) = (- 135% 594t'-455te)/1 152
In the last two equations is gL*Ten y 9.3.38 and
I v3(t)=(-42525t3+4 51737t'-8 83575P
9.3.39 below. 4-4 75475t9)/4 14720
9.3.14
vk(t)=uk(t)+t(t2-1){ ~uk-l(t)+tu~-l(t)}
Debye's Asymptotic Expansions (k=1, 2, . . .)
(ii) If p is fixed, O<r
<$
p and v is large and
(i) If a is fixed and positive and v is large and positive
positive
9.3.7
9.3.8
Y,(v sech a)- 9.3.17
= I - 81 cot28+462 cot' 84-385 cot6B+ . . .
where 11529
13. BESSEL FUNCTIONS OF INTEGER ORDER 367
9.3.18 9.3.26
{{
-3 cot ~ + cot3 B- . . .
5 17 1
24v gl(z)=-- 70 z3+-
70
Also
549 z8-- 110767 z5+- 79
g3(~)=- z2
28000 693000 12375
9.3.20 The corresponding expansions for (Y+ z ~ ~ / ~ )
~ : ( v sec 0)=.J(sin 20)/(rv){ 8) cos P
N(v, and H!a)(v+~v1/3) obtained by use of 9.1.3
are
-O(Y, 8) sin e} and 9.1.4; they are valid for --)r<arg v<#r and
where -#?r<arg v<+r, respectively.
9.3.21 9.3.27
--y2/3
22/3 hk(z)
J:(v+zv~/~)Ai' (-2ll3z) 1 )
+c
-
(ID
k=l V
=1+ 135 cot2 /3+594 cot' 84-455 cot8B - . . .
11529
{
9.3.22
}-
9.3.28
Y:(v+ zV1I3) - 2213
y/
23 Bi' (-2ll3z) 1+CV
k-1
hdz)
Asymptotic Expansions in the Transition Regions
When z is fixed, IvI is large and larg v
b
I
<
9.3.23 where
21/3
{1+2}
J . ( ~ + z v ' / ~ ) - Ai (-21/3~)
~
fk(z>
-p OD
9.3.29
+Ai' (-2%)
v
22/3 gk(z)
-
k;IO VZkl3
9.3.24 57
-&I
=) z5+- 22
100 70
Y , ( V + Z V ~ 21~ Bi - ~
~ 3 ) - (-2ll3z)
1 l+C---)
fk(z)
OD
k-1 V*I3 699 2617 23
h3(~)=- z8-- z3+-
3500 3150 3150
22/3
-- Bit (-21/3z)
V
2 &@
k-0 Val3 27 46631 $+-
3889 z4-- 1159
h4(z)=- z"--
20000 147000 4620 115500
where
9.3.25 9.3.30
1 3 1
jl(Z) =-- z I0(z) =5 z3-%
5
j*(z)=-i@
3
35
z6+-22 I, (2) =-- z4+l
13'
140 5
j3(z)=-957 z8- -173 z3-- 1 9 28+- 5437 z5-- 593 z2
7000 3150 225
&(z) =-500-
4500 3150
j ( )-- 27 z10-- 23573 z7+- 5903 z4+- 947 369 999443 31727 947
4 2 - &(z)=- z9-- z8+- z3+-
20000 147000 138600 346500 7000 693000 173250 346500
14. 368
9.3.34 Y:(v) - 1 +c
3ll2b
7 k=l 5{
'Yk
v
BESSEL FUNCTIONS IF INTEGER ORDER
9.3.37
+e2ri13Ai v5/3
(e2ri13v2f3
I
When v++ m , these expansions hold uniformly
sponding expansion for H?'(vz) is obtained by
I -%
with respect to z in the sector larg z]5 ?r- e, where
e is an arbitrary positive number. The corre-
k=O
bdl)
V*
{
where changing the sign of i in 9.3.37.
Here
21
13
a=-=.44730 73184, 3+a=.77475 90021 9.3.38
321317($1
21
23
b=--.41085 01939, 3*b=.71161 34101
31f3r(g)-
ffo=l, cy1=--=- .004, equivalently,
225
~r2=.000693735 . . ., ~~3=-.00035 38 . . +
9.3.39
1
Bo=7q=.O1428 57143. . .,
1213 =-.00118 48596.. ., the branches being chosen so that is real when
'l=-10 23750
z is positive. The coefficients are given by
&=.00043 78 . . ., &=-.OOO38 . . .
23
*/o=I, 71=-=.00730 15873 . 1 .,
3150
yz=- .00093 7300 . . ., 73= .00044 40 . . .
,5' ,6 1 - 947 = -
~
.00273 30447 . . .,
3 46500
62= .00060 47 . . . , 63= -.00038 . . .
Uniform Asymptotic Expansions
These are more powerful than the previous ex- 9.3.41
pansions of this section, save for 9.3.31 and 9.3.32, 6sfl
(284-1) (2~+3) . . (68-1)
.
but their coefficients are more complicated. They ha=
s! (144)' 9 p,=--
6s- 1 A,
reduce to 9.3.31 and 9.3.32 when the argument
equals the order. Thus a,,({) 1,
=
9.3.35
9.3.42
5 1 5 1
Ai'(v2I3{) bk({)
bo({) =-=+? 124(1 -z2)3/2-8(1 -z2)i 1
+ v5/3 27 1 =-- 5 1 5 1
48l2+(-s)i '24(z2- 1)312+8(~2- 1
1))
9.3.36
Tables of the early coefficients are given below.
For more extensive tables of the coefficients and
for bounds on the remainder terms in 9.3.35 and
9.3.36 see t9.381.
15. 369
{-
BESSEL FUNCTIONS OF INTEGER ORDER
Uniform Expansions of the Derivatives
With the conditions of the preceding subsection
9.3.4!3
Co({)"-pj 1 p+.146{-', d,({)=.OO3.
For {<-lo use
1
bo(t))"z r2, a1(f)=.000,
CO@) " --pj -1.33 (-f) -5/2, d, ({)= .OOO .
Maximum values of higher coefficients:
I bl ({)1 = .OO3, J u ~ = .0008,
({) I Id2 ({)I = .001
Icl({)I=.008 ({<lo), C1({)--.003{* as {++OD.
9.4. Polynomial Approximations
where
9.4.1 -31x13
9.3.46
2kS1 J~(x)
= 1-2.24999 97(~/3)'+1.26561 0 8 ( ~ / 3 ) ~
ck({)=-r* C
8=0
(1--2)-tl
~~{-~~/~v2+~+1(
-.31638 66(~/3)~+.04444
79(~/3)'
2k
d&) =E ia{-3sflv2k-x{ -z2)-t}
(1 -.00394 44(~/3)'~+.000210 ( ~ / 3 ) ~ ~ + ~
0
=O 8
5
1
tl
X
< 10-8
and vk is given by 9.3.13 and 9.3.14. For bounds
on the remainder terms in 9.3.43 and 9.3.44 see 9.4.2 0<x13
[9.381.
Yo(x)= ( 2 / ~ln($x)J,(x) + .36746 691
)
1 a
1 (r) Gd-r) di (r) +.60559 366(~/3)'-.74350 3 8 4 ( ~ / 3 ) ~
+ .25300 1 17 (2/3)'- .04261 2 14(~/3)
0 0.0180 -0. 004 0. 1587 0. 007
1 .0278 -. 004 . 1785 . 009 + .00427 916(~/3)"- .00024 846 ( ~ / 3 )
12+ t
2 . 0351 -. 001 . 1862 .007
3
4
. 0366 +. 002
.0352 .003
. 1927
.2031
.005
.004
<ltI 1.4X lo-*
5 .0331 . 004 . 2155 .003
6 .0311 . 004 . 2284 . 003 9.4.3 31x< 03
7 . 0294 .004 . 2413 .003
8 . 0278
. 0265
.004
. 004
. 2539
. 2662
. 003
. 003
Jo(x)=x-tfo cos e, Yo(x)=x-+j, sin e,
9
10 .0253 .004 . 2781 . 003
fo=.79788 456- .OOOOO 077(3/~) .00552 740(3/~)'
-
-.00009 512(3/~)'+.00137 237(3/~)'
--r bo(r) a1 (I) d-r)
-.00072 805(3/~)~+
.00014 476(3/~)*+t
0 0.0180 -0.004 0. 1587 0. 007
1 .0109 -. 003 . 1323 .004 [el< 1.6X lo-*
2 . 0067 -. 002 . 1087 . 002
3 . 0044 -. 001 . 0903 . 001 2 Equations 9.4.1 to 9.4.6 and 9.8.1 to 9.8.8 are taken
4 . 0031 -. 001 .0764 . 001 from E. E. Allen, Analytical approximations, Math. Tables
5 .0022 -. 000 . 0658 . 000 Aids Comp. 8, 240-241 (1954), and Polynomial approxi-
6 . 0017 -. 000 . 0576 . 000
7 . 0013 -. 000 . 0511 . 000 mations to some modified Bessel functions, Math. Tables
8 fool1 -. 000 .0459 . 000 Aids Comp. 10, 162-164 (1956) (with permission). They
9 .0009 -. 000 . 0415 . 000 were checked at the National Physical Laboratory by
10 .0007 -. 000 . 0379 .om systematic tabulation; new bounds for the errors, e, given
here were obtained as a result.
16. 370 BESSEL FUNCTIONS OF INTEGER ORDER
e0=Z- .78539 816- .04166 397(3/4
- *t
J,.,~,y v . a
J,.,~, and Y:,a respectively, except that z=O
-.00003 954(3/~)~+.00262 573(3/~)~ is counted as the h t zero of J ( ) Since Az.
Ji(z)=-Jl(z), it follows that
-.00054 125(3/~)’-.00029 333(3/~)‘
. *I
+.00013 558(3/x)’+a 9.5.1 jL,i=O, ~o.s=ji.s-i (s=2, 3, . .)
l 17
€<X 10-8 The zeros interlace according to the inequalities
9.5.2
9.4.4 -35x53
~.,l<~.+l,l<j.,Z<j~+,,Z<jU,3< * * .
+~
z-’J (z)=*- A6249 985( ~ / 3 ) .21093 573(~/3)’
1
Y*.l<Y~+l.l<Y..2<Yv+l.2<Yu.3< * -
-.03954 289(~/3)’+
.00443 319(~/3)’
-.00031 761 (~/3)”+
.00001 109(~/3)’~+€ &j:, Y
I,
<. Y
<,:
I ,<A,
~<j., 2
lt1<1.3X10-8 < ,.
Y , :Y<Z 2<jp, 2< j;,<3 . . .
The positive zeros of any two real distinct cylinder
9.4.5 Ox 3
<5
functions of the same order are interlaced, as are
zYl(s)=(2/?r)xln(~s)Jl(z)-.6366198 the positive zeros of any real cylinder function
%‘,(z), defined as in 9.1.27, and the contiguous
+ .22120 91 (~/3)~+2.16827 09(x/3)’
function V,+,(Z).
-1.31648 27(~/3)’+ .31239 51 (2/3)’ If pv is a zero of the cylinder function
-.04009 76(~/3)”+ .00278 73(Z/3)12+c
9.5.3 Vp(z) J,(z) cos(d) + Y ( )
= , z sin(?rt)
/al<l.lXlO-’ where t is a parameter, then
9.4.6 3 Is<
9.5.4 =
u:(P,)=u.-l(P.)- U . + , ( P . >
J1(z)=s-+jl
COS e,, Yl(x)=x-+jl sin e, If u. is a zero of W;(z) then
fi=.79788 456+.00000 156(3/~)+.01659 667(3/~)~
+.00017 105(3/~)~-.00249 511(3/~)’
+.00113 653(3/~)~- .00020 033(3/~)’+~ The parameter t may be regarded as a continuous
I 1X
C4< 10-8 variable and pr, u, as functions p . ( t ) , u,(t) of t. If
these functions are fixed by
e1=~-2.356i9 449 + .12499 612 (3/4
9.5.6 p,(O)=O, u.(o)=j;, 1
+ .00005 650(3/~)’-.00637 879(3/~)~
then
+ .00074 348(3/~)‘+.00079 824(3/x)‘
-.00029 166(3/~)’++t 9.5.7
j”,*=P,(s), Y.,I=Pu(s-3) (s=l,2, . .I
lt1<9X10-8
9.5.8
For expansions of Jo(s), Yo(s>,Jl(z),and Yl(x)
in series of Chebyshev polynomials for the ranges ji,a=gv(s-1), y;.a=Cv(s-$) (s=l, 2, *)
05s<S and 0<8/z5l, see t9.371.
9.5. Zeros
Real Zeros Infinite Products
When Y is real, the functions J,(z),Jl(z), Y,(z)
and Y:(z) each have an infinite number -of real
zeros, all of which are simple with the possible
exception of z=O. For non-negative Y the 6th
positive zeros of these functions are denoted by
17. BESSEL FUNCTIONS OF INTEGER ORDER 371
McMahon's Expansions for Large Zeros
When v is fixed, s>>v and p=45
-64(p-l) (69494-1 53855p2+15 85743~-62 77237)- . . .
105(8b)7
where P=(s+$v-4)a forjv,s,P=(s+$v-$)r for yI,*. With p=(t+4v-t)al the right of 9.5.12 is the
asymptotic expansion of pY(t) large t.
for
-64(6949p4+2 9 6 4 9 2 ~ ~ - 48002p2+74 14380~-58 53627)- .
12 . .
105(88')'
where S'=(s+$v-q)a for jL.81 B'=(s+#v--))a for yl,,,B'=(t+$v+t)a for uI(t). For higher terms in
9.5.12 and 9.5.13 see [9.4] or [9.40].
Asymptotic Expansions of Zeros
and Associated Values for Large Orders Uniform Asymptotic Expansions of Zeros and
Associated Values for Large Orders
9.5.14
-
jy,l v+ 1.85575 71v1l3+1.03315 O V - " ~
-.oo397v-'-.0908v-~/3+.043v-7/3+ . . .
9.5.22 j,.,-vz(r)+C OD
k=l
fk(r> with { = ~ - ~ / ~ a ,
9.5.15 9.5.23
yv,1-v+.93157 68vlf3+.26O35lv-li3
+ .01198v-'- .0060~-'~~- .001~-"~+ . .
.
9.5.16 with { = ~ - ~ / ~ a ,
j:.,-v+ 30861 65~'/~+.07249
Ov-lf3
-.05097~-'+ .0094~-"~+ . .
. (D
9.5.17 9.5.24 j;,,-vz({)+C With { = ~ - ~ / ~ a :
b-1 P-'
y, 1.~~+1.82109
: 80~'/~+.94000 7v-l" 9.5.25
-.05808v-'- .0540~-'~~+ . . .
9.5.18
JL(jv,l)
---1.11310 28~-~/~/(1+1.48460 - ~ / ~
6 ~
+ .43294V-4/3-.1943v-2+ .019v-8/3+ . . . ) where a,, a: are the sth negative zeros of Ai@),
Ai'(z) (see 10.4), z=z(T) is the inverse function
9.5.19 defined implicitly by 9.3.39, and
Y:(yv, m.95554 8 6 ~ - ~ / ~ / .74526 lv-2/3
1) (1+
+.10910v-'~3-.0185v-2-.003v-*/~+ . . . ) 9.5.26
h(O=I4t/(1--z2)It
9.5.20
JP(jL,1)
m.67488 51~-'/~(1--.16172 ~ - ~ / ~
3 jl(r) =Mr)Ih(r) 12bo(l)
+ .02918~-~'~-.0068~-~+ . . )
. m(n=3r-'z(r>{h(r)12co(r)
9.5.21 where bo({), co({) appear in 9.3.42 and 9.3.%.
Y : J m.57319 40~-"~(1-
&, .36422 O V - ~ / ~ Tables of the leading coefficients follow. More ex-
+.09077~-*~+.0237v-~+ . . ). tensive tables are given in [9.40].
Corresponding expansions for s=2, 3 are given The expansions of yv. YXyv, y:. Iand Y ( :J
a, a), .Y.
in [9.40]. These expansions become progressively corresponding to 9.5.22 to 9.5.25 are obtained by
weaker as s increases; those which follow do not changing the symbols j, J, Ai, Ai', a, and a: to
suffer from this defect. y, Y, -Bi, -B?, b, and b: respectively.
18. 372 BESSEL FUNCTIONS OF INTEQER ORDER
-i- h (i-) fl (i-)
0. 0 1.000000 1. 25992
-. ~ _ . . 0. 0143 -0.007 -0. 1260 -0.010 0. 000
0. 2 1. 166284 1.22076 -. 0142 -. 005 -. 1335 -. 010 .002
0. 4 1.347557 1. 18337 .0139 -. 004 -. 1399 -. 009 .004
0. 6 1.543615 1. 14780 .0135 -. 003 -. 1453 -. 009 .005
0. 8 1.754187 1. 11409 .0131 -. 003 -. 1498 -. 008 .006
1. 0 1.978963 1.08220 0.0126 -0.002 -0. 1533 -0. 008 0.006
-r
-
1. 0 1.978963 1.08220 0.0120 -0.002 -0. 1533 -0.008 0.006
1. 2 2. 217607 1. 05208 .0121 -. 002 -. 1301 -. 004 .004
1. 4 2. 469770 1.02367 .0115 -. 001 -. 1130 -. 002 .003
1. 6 2. 735103 0.99687 . 0110 -. 001 -. 0998 -. 001 .002
1. 8 3.013256 .97159 .0105 -. 001 -. 0893 -. 001 .002
2. 0
2. 2
3.303889
3. 606673
0.94775
. 92524
0.0100
.0095
-0.001
-0.001
-0.0807
-. 0734
I -0.001 0.001
.001
I
2. 4 3. 921292 .90397 .0091 -. 0673 . 001
2. 6 4. 247441 .a8387 .0086 -. 0619 .001
2. 8 4.584833 . 86484 .0082 -. 0573 0.001
-
3. 0 4. 933192 0.84681 0.0078 -0.0533
3. 2 5. 292257 .a2972 .0075 -. 0497
3. 4 5.661780 . 81348 .0071 -. 0464
3. 6 6.041525 . 79806 .0068 -. 0436
3. 8 6. 431269 .78338 .0065 -. 0410
4.0 6. 830800 0. 76939 0.0062 -0.0386
4.2 7. 239917 .75605 .0060 -. 0365
4.4 7. 658427 . 74332 .0057 -. 0345
4.6 8.086150 . 73115 .0055 -. 0328
4.8 8. 522912 .71951 .0052 -. 0311
5. 0 8.968548 0. 70836 0.0050 -0.0296
5. 2 9.422900 .69768 .0048 -. 0282
5. 4 9.885820 . 68742 .0047 -. 0270
5. 6 10.357162 . 67758 .0045 -. 0258
5. 8 10.836791 . 66811 .0043 -. 0246
6. 0 11.324575 0. 65901 0.0042 -0.0236
6. 2 11. 820388 .65024 .0040 -. 0227
6. 4 12. 324111 .64180 .0039 -. 0218
6. 6 12. 835627 .63366 .0037 -. 0209
6. 8 13.354826 .62580 .0036 -. 0201
7. 0 13.881601 0.61821 0.0035 -0.0194
Complex Zeros of J,(s)
(--f)W) 81(3)
When v> -1 the zeros of J,(z)are all real. If
0. 40 1.528915 1.62026 0.0040 -0.0224 v<-1 and v is not an integer the number of com-
.35 1. 541532 1.65351 .0029 -. 0158 plex zeros of J,(z) is twice the integer part of
.30 1. 551741 1. 68067 .0020 -. 0104
.25 1.559490 1. 70146 . 0012 -. 0062 (-v); if the integer part of (-v) is odd two of
.20 1.564907 1.71607 .0006 -. 0033 these zeros lie on the imaginary axis.
0. 15 1.568285 1.72523 0.0003 -0.0014 If v20, all zeros of J ( ) are real.
Lz
. 10 1.570048 1.73002 . 0001 -. 0004
.05 1. 570703 1.73180 . m o o -. 0001
. 00 1.570796 1. 73205 . w o o -. 0000
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.
19. I
a=m=.66274 . . . FIGURE 9.6. Zeros ofHi’)(z)and Hi”’(z) . .
.
b = + J m 2=.19146 . . .
I n larg zl<?r.
and b=1.19968 . . . is the positive root of coth t The asymptote Of the solitary infinite curve is
=t. There are n zeros near each of these curves. given bY
Asymptotic expansions of these zeros for large n Y~=-+In2=-.34657 . . .
20.
21. BESSEL FUNCTIONS OF INTEGER ORDER 375
9.6.5
Y,(zeW) =et(,+l)riI ,(z1 -(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
I: When v is fked and z+O
9.6.7
FIGURE e-zlo(z),e-zIl(~),eZKO(;C)e"Kl(z).
9.8. and
Iv(+(iz)yr(v+i) (vz-1, -2, . . .)
9.6.8 Ko(z)--ln z
9.6.9 K,(z)-+r(V)(~Z)-' (gv>o)
Ascending Series
1,(2)=(42)v 2 myv+k+i)
o (42")"
9.6.10
9.6.11
Kn(z>=&(34-" go k!
(n-k-l)!
n-1
(-322))"
+ (-In+1 In ( 3 4 I n ( ~ )
+(->"3(3d" (tz")"
2 INC+l)+W+k+l) 1 k!(n+k)!
k-0
where +(n) given by 6.3.2.
is
FIGURE 1,(5) and K,(5).
9.9.
4z2 (2!)2
(1!)2 (+z">"
9-6-12 Io(~)=l+-+-+- (tz2)3+*.
(3!)2
.
Relations Between Solutions 9.6.13
9.6.2 K ( z ) = h I-,(z) -I,(z)
sin (y.) Ko(z)= - {h (3Z)+YI~O(Z) +m
4 z2
The right of this equation is replaced by its (4z">" (tz"3+*
+(1+3) (,!),+(1+3++)
limiting value if v is an integer or zero. (3!)2 a -
9.6.3
I,(z) =e-+prfJ,(zetrf) (-r<arg 2<34 Wronskians
9.6.14
I,(z) =e3fl'/2J,(=-3"/2 1 (3*<arg z 54
9.6.4
W{ I)
&, )
&I 1 =I,(z)l-~,+l~(z)-I,+l(z)I-,(z)
}
= -2 sin (vr)/(~z)
K,(z)=)riet"'H~')(zet"') (-r<arg z<$r) 9.6.13
K,(z)= -3rie-+*f HP)(ze-+")(- &<arg z <r) W{ ,K,(z) I,( z) =
I
&) + (2)
K,+l(z) Iv+l ZJ,z) = l/z
22. 376 BESSEL FUNCTIONS OF INTEGER ORDER
Integral Representations eurfKvany linear combination of
%”, denotes I”, or
9.6.16 these functions, the coefficients in which are
Io(z)=’S‘
‘ A 0 independent of z and v.
9.6.17 K~(z)=-- {?+In (22 sin2e)}&
9.6.27 I ( )
; Z = rl(z), K (2)= -K~
; (z)
9.6.18 Formulas for Derivatives
9.6.28
Ko(z) =l 0
cos (z sinh t ) d t = l mc
* dt Analytic Continuation
(X>O) =em”’‘Iu(z) (m an integer)
9.6.30 Iu(zemrf)
9.6.22
9.6.31
K.(z)=sec (3m)
l- cos (z sinh t) cosh (vt)dt
Kv( t) =e-mmf Kv(z)--?ri (mvn) csc (v?r)I,(z)
sin
=csc ( l-
3 ~ ) sin (zsinh t) sinh (vt)dt
9.6.32 I.(Z)=I.(z),
- -
K,.(B)=K,(z)
(m an integer)
(V real)
( 9<11 z>O)
14
9.6.23
Generating Function and Associated Series
9.6.33 2 tkIk(z)
k=-m
(t#O)
m
9.6.34 ez cOse=Io(z) C Ik(z)
+2 cos(k0)
k-1
9.6.24 K.(z)= cosh (ut)& (larg 2 <h)
1
J O
9.6.25
2
+2 k=l ( - ) ~ ~ ~ ( cOs(2ke)
z)
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 ) + 2 1 ~ ( ~ ) + 2 +216(2)+ . . .
1,(~)
9.6.40 sinh 2=211(2)+213(z)+21~(2)+ . .
.
*See page 11.
23. BESSEL FUNCTIONS OF INTEGER ORDER 377
Other Werential Equations
9.6.50 E m {v-pe-p"
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 For the definition of P;" and Qf, see chapter 8.
time the symbol W in the given solutions is
replaced by 3 . Multiplication Theorems
9.6.51
9.6.41
zzw" + z( 1 f 22) 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
Zeros
9.6.46 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.
9.6.46 For example, if v is real the zeros of IJz) are all
complex unlese -kv-
2<< (2k- 1) for some posi-
tive integer k, in which event I,(z) has two real
zeros.
Expreesions in Terms of Hypergeometric Functions The approximate distribution of the zeros of
9.6.47 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 z <$a; this result remains true when n is
I
replaced by any real number v.
9.6.48
9.7. Asymptotic Expansions
OF^ is the generalized hypergeometric function. Asymptotic Expansions for Large Arguments
For M(a, b, z), Mo,,(z) Wo,,(z) see chapter 13.)
and When v is fixed, IzJis large and r=49
Connection With Legendre Functions
9.7.1
If LL and z are fixed, Wz>O, and v+w through
real positive values
9-6-49 l {r
m e
i v
