The document discusses spherical Bessel functions of fractional order. It defines the spherical Bessel functions of the first kind jn(z), the second kind yn(z), and the third kind hn(z). It provides representations of these functions by elementary functions, ascending series, Poisson's integral formula, and Gegenbauer's generalization. It also discusses properties such as differentiation formulae, analytic continuation, and generating functions. Tables are provided with values of the modified spherical Bessel functions for different orders and arguments.

2. 436 BEBBEL FUNCTIONS OF FRACTIONAL ORDEB
page
Table 10.8. Modified Spherical Beasel Functions-ordera 0, 1 and 2
(01255). . . . . . . . . . . . . . . . . . . . . . . . . . . 469
&&In+* (5), 4ZG-i
(z>
n=O, 1, 2; ~=0(.1)5, 4-9D
Table 10.9. Modified Spherical Beasel Functions--Orders 9 and 10
(O<zl.p). . . . . . . . . . . . . . . . . . . . . . . . . . . 470
d 3 * l Z I n + i (2) xn+'J+GEKn+i (2)
~ = 9 10; ~=0(.1)5, 7-8s
,
e-'I* (4,(2/TWKn+i (z)
n=9, 10; z=5(.1)10, 6 s
6 exp [-z+n(n+ 1)/(2z)IIn~(z)
exp [z-n(n+1)/(2z)l~n+i(z)
n=9, 10; z-'=.1(-.005)0, 7-8s
. Table 10.10. M~dified Spherical Beasel Functione-Various Orders
(OIn5100) . . . . . . . . . . . . . . . . . . . . . . . . . 473
&Wn+l(z) , 45%+i (z)
n=0(1)20, 30, 40, 50, 100
z=1, 2, 5, 10, 50, 100, 10s
. Table 10.11. Airy Functions ( 0 5 2 1 -) . . . . . . . . . . . . . . 475
Ai&), Ai'(z), Bi(z), Bi'(z)
z=O(.O1)1, 8D
Ai(-z), Ai'(-z), Bi(-z), Bi'(-z)
z=O(.O1)1(.l)lO, 8D
Auxiliary Functions for Large Positive Arguments
Ai(z)=&z-%-Y(-€); Bi(z)=z-l/rdf([)
Ai'(z)=-42Y4e-(g(-€); Bi'(z)=z'/Wg(€)
c'=
f(f I),g(f [) ; )=&I+/', 1.5(- .l).5(- .05)0, 6D
Auxiliary Functions for Large Negative A r p e n t a
Ai(+ =z-"*lfi(O COB €+jd€) sin €1
Bi(-z)=z-l~'ffa(€) €-fi(€) sin €1
Ai'(-~)=z'/~[g~([) €-g2(€) cos €1
sin
Bi'(-z)=z'/Ygal€) sin €+gl(€)COB €1
f1(€)J f'(€1 9 9 (€) 9 (€1 ; €=
1 9 2 j2""
p = . 0 5 (-.01)0, 6-7D
Table 10.12. Integrals of Airy Functions (OIz110). . . . . . . . . 478
Ai(t)dt, z=0(.1)7.5;
SD': Ai(--t)dt, z=O(.l)lO, 7D
s,'Bi(t)dt; 2=0(.1)2; 1 Bi(--t)d, z=0(.1)10, 7D
Table 10.13. Zeros and Associated Valuea of Airy Functions and Their
Derivatives (1 1 8 1lo) . . . . . . . . . . . . . . . . . . . . 478
Zeroa a,, a;, b,, b; of Ai@), Ai'@), Bi(z), Bi'(z) and valuea of Ai'@,),
&(a;),Bi'(b,), Bi(b;) e=l(l)lO, 8D
Complex Zmos and Associated Valuea af Bi(z) and Bi'(z)
(1 5 8 5 5 )
Modulus and Phaae of Bi'(@,), Bi(8:) 8=1(1)5, 3D
The author acknowledge8 the eseietance of Bertha H. Walter and Ruth Zucker in the
preparation and checking of the tablea and graphs.

3. {
10. Bessel Functions of Fractional Order
Mathematical Properties
10.1. Spherical Beeeel Functions
Definitions
Werential Equation
10.1.1
$ " +2m'+[ 2-n(n+ 1) ]w=O
0
(n=O,*l, f 2 , . . .) Reprementotiomby Elementary Functions
Particular solutions are the Spherical Bessel 10.1.8
functions of thefist kind
j,(z)=z-'[P(n+*, 2) sin (z-+W)
jm(z>= W d n + i ( z ) +Q(n+h 4 COS (z-hll
the Spherical Bessel functions o the second kind
f 10.1.9
yn(Z)=(-l)m+lz-'[P(n+), Z) COS (Z+#TJT)
Ya(z) =AGYn++(z) ,
and the Spherical Bessel functions of the th&d -Q(n+3, Z) sin ( z + h ) l
kind
hP(z)=jn(z) +iya(z)=dGFHSt(z),
hia)(z>=jm(z)-iyn(z) = ~ Z H ; ~ + ( Z > .
The pairs jn(z), ya(z) and hil)(z), hf)(z) are
linearly independent solutions for every n. For
general properties s the remarks after 911
m ...
-9(-l)&(n+$, 2k)(22)-*
0
Ancending S e r i e a (&e 9.1.2.9.1.10)
10.1.2
2s 32'
'm(z)=l. 3 . 5 . . . (2n+1) '-1!(2n+3)
1..
013
="y 2k+1)(22)-"-'
(-l)&(n+*,
0
163.5.. . (2n-1) 33 (n=O, 1,2, . . .)
Y() -
mz= .p+ 1 l! (1-24
' 2 V n (3z')'
) (3-2%) - . . .}
(n=O, 1,2, . . .)
1680
16120 I 30240
437

4. FIGUSID js(~),
10.3. va(Z). ~=10.
Pohon's Integral and Gegenbauer's Generalization
10.1.13 jn(z)==2' s,- cos (2 cos e) sin2"+'8
ds
(See 9.1.20.)
10.1.14
-.3-
=z1 ( - i ) n l e'*coaep,,(cos e) sine&
F I G U B ~ j&). n=0(1)3.
10.1. (n=O, 1,2, . . .)
*See page II.

5. BESSEL FUNCTIONS OF FEtACTIONAL ORDER 439
Spherical Beaeel Functions of the !bond and T i d
hr
Kind
10.1.15
yn(2)=(-l)"+lj-n-1(2) (n=O, f l , f 2 , . . .)
10.1.16
hz)(z)=i-m-1z-1el'& (n++,k)(-2izl-k
0
10.1.17
n * (See 9.2.28.)
hi2'(z)= i R + l z - l e - i (n++ k) (2iz)
Z~
0
10.1.28 ( X Z M , =j:(2)
+ / ) f(2)
2 +y: (2)=2-2
10.1.29
(+X/Z>M,~,~(Z) +y;(z)
=j;(z) =~-~+2-'
10.1.30
0 n-2k
(n=O, 1,2, . . .)
Analytic Continuation
DHertmtiation Formulae
10.1.34 jn(zernri) ernnrfjn(z)
=
10.1.35 yn(zernrl) (-I)"'ernnrfyn(z)
=
10.1.36 h:)(~p.(~+l)~~
)= (-l)%;s)(z)
10.1.37 h:2)(ze(h+1)rt (-l)"h:Q(Z)
)=
10.1.38 h$ (zeMr*)
=hii) (z)
(Z=l, 2; m,n=O, 1,2, . . .)
Generatiqg Functiom
10.1.39
10.1.26
*SeePage n.

6. 440 BESSEL FUNCTIONS O FRACTIONAL ORDER
F
Fresnel Integrab
10.1.53
(See also 11.1.1, 11.1.2.)
Zeros and Their Asymptotic Expansions
The zeros of j,(z) and y() are the same as the
,z
zeros of Jn+&) Y,+)(z) and the formulas for
and
j.,, and y, given in 9.5 are applicable with
.,
v=n+#. There are, however, no simple relations
connecting the zeros of the derivatives. Ac-
cordingly, we now give formulas for a;,,, b;,,, the
8-th positive zero of j: (2) , y (2), respectively;
;
z=O is counted as the first zero of jA(z).
(Tables of a;,,, b L , jn(4J,
YnK.,) are given in
[10.311.)
Elementary Relatiom
fn(z) =jn(z) d+Yn(z) sin
(t a real parameter, 0 St 5 1)
If T,, is a zero of &(z) then
Some In6nite Seriee Involving j:(z)
10.1.50 2 (2n+l)j:(z)=l
0
10.1.51
m
sin
(-1)R(2n+1)j:(~)=3--22
0
10.1.52 10.1.57

7. BESSEL FUNCTIONS O FRACTIONAL ORDER,
F 441
McMahon's Expadons for n Fixed and a Large UniformAsymptotic Expansions of Zeros and
Aeeociated Values for n Large
10.1.58
10.1.63
d . 8 ~ b:,8-8-((c(+7) (8/3)-1
4
-- (7p2+154/~+95)(80))'~
3
-32 (85/~~+3535p~+356lp+6133)(8/3)-~
15
--
64 (6949p4+474908p3
+330638p2
105
+9046780p-5075147) (8/3)-'-- . . .
@=~(s+3n-3)for /3=*(s+$n)
for
p=(2n+1)2
{
Asymptotic Expansions of Zeros and Associated Values
for n Large
10.1.59
a;. -(n+3)+ .8086165(n+ 3)1/3-.236680(n+3) -'I3
-.20736(n+$)-1+.0233(n+$)--"/3+ . . .
10.1.60
b:, -(n+ 3) + 1.821098O(n+ 3)' p
11+~~xi(n+t)-"3b:i(n+3)-~}
k=l
+.802728(n+4) -m-.11740(n+4)
h&), z(€) are defined as in 9.5.26, 9.3.38, 9.3.39.
+.0249(n+$)-6/3+ . . . a:, b s-th (negative) real zero of Ai'(z), Bi'(z)
;
10.1.61 (see 10.4.95, 10.4.99.)
Complex Zeros of h!"(z), h!"'(z)
j,(a:, ') -.8458430(n+$) 1-
4'" .566032(n+3) -2/3
hil)(z) and hi')(z.em**), m any integer, have the
+.38081(n+3)-4@-.2203(n+3)-a+ . . .} same zeros.
hi1)(z) has nzeros, symmetrically distributed with
10.1.62 respect to the imaginary axis and lying approxi-
}
mately on the finite arc joining z=-n and z=n
&(b:, 1) 4 1 8 3 9 2 1 (n+4)-"'"{ 1-1 .274769(n+4))-2/3 shown in Figure 9.6. If n is odd, one zero lies on
+1.23038 (n+ 3) -4/3- 1.0070 (n+4)-2+ . . . the imaginary axis.
hi*)'(z)has n + 1 zeros lying approximately on the
See [10.31] for corresponding expansions for same curve. If n is even, one zero le on the
is
8=2, 3. imaginary axis.

8. BESSEL FUNCTIONS OF FRACTIONAL ORDER
-z (-mi (r) (-I)'& (z)
0. 0 -. 4409724
-. 4572444
-. 122500
-. 114201
-. 06806
-. 05986
. 000000
.027518
. 00000
.00575
.oooo
0. 2 .0023
0. 4 -. 4702250 -. 10'1243 -. 05279 .049069 .01118 .0043
0. 6 -. 4802184 -. 101318 -. 04674 .065677 .01592 .0061
0. 8 -. 4875705 -. 096159 -. 04160 .078255 .01983 .0075
1. 0 -. 4926355 -. 091561 -. 03725 .OS7587 .02290 .0085
--3 hi (r) Hi (r)
1. 0 -. 4926355 -. 09156 -. 037 .OS7587 .0229
1. 2 -. 4131280 -. 05056 -. 014 .065507 .0121
1. 4 -. 3551700 -. 03043 -. 006 .050524 .0070
1. 6 -. 3108548 -. 01950 -. 003 .039890 .0042
1. 8 -. 2757704 -. 01310 -. 001 .032085 .0027
2. 0 -. 2472521 -. 00914 .026206 .0018
2. 2 -. 2235898 -. 00658 .021682 .0012
2. 4 -. 2036314 -. 00485 .018141 .OW8
2. 6 -. 1865701 -. 00366 .015326 .0006
2. 8 -. 1718217 -. 00280 .013061 .0004
3. 0 -. 1589519 -. 00219 .011217 .0003
3. 2 -. 1476304 -. 00173 .009701 .0002
3. 4 -. 1376005 -. 00138 .008443 .0002
3. 6 -. 1286601 -. 00112 .007391 . 0001
3. 8 -. 1206469 -. 00091 .006505 .om1
4.0 -. 1134296 -. 00075 .005753
4.2 -. 1069004 -. 00062 .005111
4.4 -. 1009699 -. 00052 .004560
4.6 -. 0955634 -. 00044 .004085
4.8 -. 0906180 -. 00037 .003672
5. 0 -. 0860804 -. 00032 .003313
6. 2 -. 0819049 -. 00027 .002998
5. 4 -.0780523 -. 00023 .002722
5. 6 -. 0744888 -. 00020 .002478
5. 8 -. 0711850 -. 00018 .002262
6. 0 -. 0681152 -. 00015 .002070
6. 2 -. 0652570 -. 00013 .001899
6. 4 -.0625905 -. 00012 .001746
6. 6 -. 0600985 -. 00010 .001609
6. 8 -. 0577653 -. 00009 .001486
7. 0 -. 0555773 -. 00008 .001375
0. 40 -. 0645731 -. 00013 .001859
.36 -. 0487592 --. 00005 .001056
.32 -. 0352949 -. 00002 .OW551
.28 -. 0242415 -. 00001 .000259
.24 -. 0155683 .000106
.20 -. 0091416 . oooo37
. 16 -. . 0047276
- -. -
. .000010
. 12 -. 0020068 .000002
.08 -. 0005965
.04 -. 0000747
.oo I -.0000000 I I

9. BESSEL F " I 0 N S OF FRAcTlONAL ORDER 443
10.2. Modified Spherical Bessel Functions
Definitions
Differential Equation
10.2.1
1)
z2w" +2zw' - z2+n(n+ ]w=0
[
(n=O, f l , f 2 , . . .)
Particular solutions are the Modifid Spherka?
Bessel fum!;.nS o the$rst kind,
f
10.2.2
{
&$&++(z) = e-nri/*jn(zer*/a) (-r<arg z 5%.>
= e3nrf/2jn(~e-3rt/a) (I)lr<arg z 27)
o the second kind,
f
10.2.3
-I- ,++ (z)=ea(n+l)rt/ (2e
(-r<arg 2 5 31r)
-e-
- (n+1)rt/%~(~~-3rt/a)
(h<arg z5r)
o the third kind,
f
10.2.4 (n=O, 1,2, . . .)
(See 10.1.9.)
r n K + + ( Z =fr(-l)"+'dm[In+t(z) -I--)I-+(Z)l
)
10.2.12
The pairs
WzIn++(z),n Z I - n - + ( Z )
r &&zI,++(z)=g,(z) sinh z+g-n-l(z) cosh z
and go(2) =2-1, g1(z) = -2-2
dGmn++(Z) 9 rnZKn++(Z>
gn-1(4 -gn+1(z) = (2n+1) z-'gn(z>
are linearly independent solutions for every n. (n=O, f l , f 2 , . . .)
Most properties of the Modified Spherical
Bessel functions can be derived from those of the The Functions 1/&&2I*cn++,(Z), n=O, 1, 2 .
Spherical Bessel functions by use of the above
relations. 10.2.13
=z
Ascending %rim sinh
10.2.5 rnIl,2(Z> 2
sinh 2-- 3 cosh z
Z2
10.2.14
10.2.6 cosh z
1 . 3 . 5 . . . (2n-1) MZI-l/2(Z) =- Z
4GI-n-+(Z)= (-1)"2"+1
sinh z cosh z
-3,2 (2) =-
- -
3za +
&
+I
Z 22
'+1!(1-2n) 2!(1-%)(3-%) 3
(n=O, 1,2, . . .) --
I-
&
&)
&
=
22
sinh z+
f
*See page 11.

10. 444 BESSEL FUNCTIONS OF FRACTIONAL ORDER
Modified Spherical Besee1 Functions of the T i d Kind
hr
10.2.15
&&zKn++(z)
=3?rie"+1)~"2h~~)(ze*'f)
(-*<wg z 5.
31
--3,,.ie-
- (n+1)ri/2h(2)(ze-+rt)
I
O,,.<arg 2 54
=($?r/z)e-Z 5 (n++,k)(22)-~
0
10.2.16
K,+*(z)=K-~-~(z)(n=O, 1 , 2 , . . .)
The Functions m R n + + ( ~ ) , n = O ,2
1,
10.2.17 &@&(z)= (h/z)e-'
&?2GI2(z) = ($n/z)e-Yl+z-')
mK~,2(Z)=(3?r/Z)e-'(l+32-'+32-e)
Elementary Properties
Recurrence Relations
f&): r n L + * ( Z ) ,
(-l)"+'mKn++(Z)
(n=O, f l , &2, . . .)
10.2.18 fn-l(z) -fn+1(2) = (2n4-1)z-'fn(z)
d
(z>=@n+1) fn(z>
10.2.19 nfn-l(z)+(n+l)fn+l s
n+l d
10.2.20 7fn(z) +&fn(4 =fn-1(z)
(See 10.2.22.)
n d
10.2.21 --fn(z)
Z
+-&fn(z) =fn+1(z)
(See 10.2.23.)
DifYerentiation Formulae
fn(2): r n L + * ( Z ) , (-w+1m&++(4
(n=O, k l , ~ 2 ,. . )
10.2.22 (; g J k + I=
'f.(z) Zn-m+'f,-,(z)
10.2.23 (5 -g>l -y n
[z (2) I =z -n-mfn+m (2)
10.5. -&&+*(z),
FIQURE &K,.++(z). z=10.
(m=1,2,3,. . . )

11. BESSEL FUNCTION8 OF FRACTIONAL ORDER 4.45
Formulas of Rayleigh's Type Addition Theorems and Degenerate Forms
T, p, 6, X arbitrary complex; R=,@+p2-2rp coa 6
10.2.35
10.2.25
(n=O,1,2, . . . )
=l
- n (.-1)k+1 (2n-k) ! (2n-2k)!
(22)%-*"
2 3 0 k! [(n-k) I]*
10.3. Riccati-Beasel Functions
DilTerential Equation
Generating Functiom 10.3.1
16.2.30 9w"+ [z5 -n(n +l)]w=0
(n=O, f l , f 2 , . . .)
Pairs of linearly independent solutions are
zjn(z), q/n(z)
(2I t I <I)
Iz
10.2.31 &p' (z), zhp (2)
All properties of these functions follow directly
from those of the Spherical Besael functions.
The Functiom zj,(z) zy&) n=O, 1, 2
Derivativea With Reapect to Order
10.3.2
10.2.32
zy'o(z)=sin 2, Zj,(z)=z-'sin 2-cos 2
zy'2(Z)=(3Z-2-1) S h 2-32' COB 2
10.3.3
10.2.33 q/&)=-cos 2, q/1(2)=-sin 2-2-1 cos 2
2&(2)=-32-' Sin 2-(32-'-1) COB 2
WTOMkiIUM
10.3.4 W{zy'n(z),zYn(z)}=l
10.3.5 W{z@)(z), zhF)(~)}
=-2i
For El(z)and Ei(z), see 5.1.1, 5.1.2. (n=O, 1,2, . . .)
*See page n.

12. 446 BESSEL FUNCTIONS OF FRACTIONAL ORDER
10.4. Airy Functions 10.4.12 W{Ai(z), Ai ( ~ e - ~ * ~ / ~ ) }
=&r-lerfl6
Definitions and Elementary Properties 10.4.13 W{Ai ( ~ 6 ~ , ~ ~ (1 e - )
Ai ~ ~ ~ *~/~)}
=4ir-1
Differential Equation
10.4.1 w" -zw=o
Pairs of linearly independent solutions are
Ai (z), Bi (z),
Ai (z), Ai ( 2 W 3 ),
Ai (z), Ai ( ~ e - ~ ~ ~ / ~ ) .
Ascending Series
10.4.2 Ai (z) =clf(z) -c2g(z)
10.4.3 Bi (z) = & [cJ(z) +c~g(z)l
1 1-4 1.4.7
f(z)=i+- z3+- z6+- zO+. . .
3! 6
! 9!
=$ 3kk);( G!
2 2.5 2.5.8
g(z)=z+-4! z4+- 7 ! z7+- lo! zl0+ . . .
23k+1
=$ 3k ( )
gk (3k+l)!
(a+$),=l
3k(a+i)k=(3sl+1)(3a+4) . . . (&+3k-2)
(a arbitrary; k=l, 2, 3, . . .)
(See 6.1.22.)
10.4.4
el=& (O)=Bi (0)/8=3-~fi/r(2/3)
=.35502 80538 87817
10.4.5
c2=-Ai' (@)=Sif(0)/fi=3-ifl/r(1/3)
=.25881 94037 92807
Relations Between Solutions
10.4.6- Bi (2)=erfnAi (a"'/"> Ai (ze-*"fl)
+e+@
10.4.7
Ai (z)+e"fn Ai (a2rtD) +e-2*iD Ai (a-2rfn)=o
10.4.8
Bi (2) +e*fn Bi (=2r*/g> +e--2rfn Bi (=-*rim) =O
(2)G
10.4.9 Ai (~e*~~'/~)=)e*'~/~[Ai Bi (z)]
Wl-OMkifUl9
10.4.10 W{Ai (z), Bi (z)}= T - ~ .."
I
~
10.4.11 W{Ai (z), Ai ( ~ e ~ * ~ =+r-1e-rf/6
/~)} FIGURE
10.7. Bi (~zz),
Bi' (*4.

13. BESSEL FUNCTIONS O F FRACTIONAL ORDER 447
10.4.30 I*2n({)=(@/2z)[f43 Ai'(z)+Bi'(z)]
10.4.31 K*2n({) ( ~ / z ) Ai' (z)
=- T
{
Integral Representations
10.4.32
(3a)-% Ai [f (3a)-lBz]=Lm (at3ffzt)dt
cos
10.4.33
=l
(3a)-lB7r Bi [f (3a)-lnz]
[exp (--at3ffzt)+sin (at3fzt)ldt
The Integrals l' Ai ( f t)dt, la Bi ( f t ) d t
=#z3/2
10.4.34 l Ai (t)dt=$ s,' [1-113(t)-111&)b!i
10.4.35 Ai (-t)dt=-
:sd [J-113(t)+J113(t)]dl
Ascending Series for is Ai ( f Qdt, is Bi ( f t)dt
Representations of Beosel Functions in Terms of Airy
Functions
10.4.38 l Ai (t)dt=clF(z)-c2G(z)
(See 10.4.2.)
10.4.39 1 Ai (-t)dt=-clF(-z)+c2G(-z)
10.4.22 5*1/3({)=)m[& Ai (-z)FBi
90.4.23 (-z)-i
H:'l,3({)=e~d/6m[Ai
(-2))
Bi (-z)]
10.4.40 1 Bi (t)dt=@[clF(z)+c2G(z)l
(See 10.4.3.)
10.4.24 H2',,,({)=efd1"m[Ai(-z)+i Bi (-z)] 10.4.41
10.4.25 1*l/a({)=$mz[r& Ai (z)+Bi (z)] s,'Bi (- t)dt= -fi[clF(- z) +c2G(- z) I
10.4.26 K*ln(l) =rm
Ai (z) 1 1.4 1.4.7
F(z)=z+- z4+-7 z7+- 1O ,lo+ . . .
10.4.27 J,tzn({)=(J3/2z)[fJ3 (-z)+Bi'(-z)]
Ai' 4
! ! !
10.4.28
Hi({)=e-2rf /3H(l)
it -21&)
=erf~'(J3/z)[Ai'
(-2)-i Bi' (-41 G(z)=B z'+-2 z~+-z'+-z~'+ 5 . 8
1 2.5 2. . .
5! 8! ll!
10.4.29 p + 2
Hi~~({)=e2rf/3H~z~1a({) =$ ":(
k
) (3k+2)!
(J3/z)[Ai' (- z) +i (- z)]
Bi' The constanta cy,c2 are given in 10.4.4, 10.4.5.
'See page n.

14. 448 BESSEL FUNCI’IONS OF FRACTIONAL ORDER
Tbe Functions Gi(z), Hi+) Difterential Equations for Gi (z), Hi (z)
10.4.42 10.4.55 w’-zw=-?r-l
Gi ( ~ ) = r - ~ ~ ~ s i n ( i t ~ + z t ) d t
1 1
w(0) =- Bi (0) =- Ai (0) = .20497 55424 78
3 6
=IBi (z)+
3 s,’[Ai(z) Bi (t)-Ai (t) Bi (z)]dt
(
1
w’(O)=-Bi’ O ) = - l Ai’(0)=.14942 94524 49
10.4.43 3 6
w(z)=Gi(z)
Gi’@)=:Bit (z)+J’[Ai’(z) (t)-Ai (t) Bi’(z)]dt
Bi
3 0
10.4.56 - w’ -zw= r-1
’
10.4.44 2
w(O)=- Bi
3
(O)=A (0)=.40995 10849 56
6
Ai
Hi(z)=s-’fomerp(-i P+zt) dt
2 2
w’(O)=-Bi’(O)=-- Ai’(0)=.29885 89048 98
=-Bi (z)+
3 s , ’
“
[Ai (t) Bi (2)-Ai(z) Bi(t)]dt 3 fl
w(z)=Hi (z)
10.4.45
2 Differential Equation for Products of Airy Functions
Hi’(z)=3Bi’
(z)+
s,’[Ai (t) Bi’(z)-Ai’(z) (t)]dt
Bi
10.4.57 w’ -4ZW’ -2w=o
“
10.4.% Gi (z)+Hi (z)=Bi (z) Linearly independent solutions are Ai2
s’
(2))
Representations of 1’ Ai( f t)&, Bi( f t)&
Ai (z) Bi (z), Bi2 (2).
Wronskian for Products of Airy Functions
by Gi (kz), (*z)
Hi
10.4.47 10.4.58 W{Ai2(2))Ai (z) Bi (2)) Bi2 (z) ]=27r3
1
b i (t) d t =-+ T[ (z)Gi (z) -Ai (z)Gi’ (z)]
Ai’ Asymptotic Expansions for Iz( Large
3
10.4.48
=--- n[Ai’(2) Hi (2)-Ai (z) Hi’(z)]
3
10.4.49 +I a=-- 6k+l ck
) k
6k-1
(k=1,2, 3, . . .)
1
b i (-t)dt=----?r[Ai’ (-2) Gi (-2)
3
-Ai (-2) Gi’(-41
10.4.50
=?+r[Ai’ (-2) Hi (-2)
3
-Ai (-z)Hi’ (- z) ]
10.4.51
1 Bi (t)dt=?r[Bi’ Gi (2)-Bi
(2) (2) Gi’(z)]
10.4.52 =-r[Bi’ (z) Hi (2)-Bi (z) Hi’(z)]
10.4.53
l* Bi (-t)dt=--?r[Bi’ (-2) Gi (-2)
-Bi (-2) Gi’(-z)]
10.4.54 =*[Bi’ (-2) Hi (-2)
-Bi (-2) Hi’(-z)]

15. BESSEL F U " I O N S OF FRACTIONAL ORDER 449
10.4.62 10.4.70
Ai' (-Z)--T-+Z* Ai' (--s)=N(z) cos +(s),Bi' (-r)=N(z) sin+(t)
N(z)=JIAi'o (--s)+Bi'2 (-s)],
+(s)=arctan [Bi' (-t)/Ai' (-2)l
Differential Equations for Modulue and Phase
Primes denote differentiation with respect to x
10.4.71 -r-l, W+'= *-Ix
-
10.4.72 jV2,M~i!+M2p=M'2 + r-2M-2 *
10.4.73 NN'=-xMM'
10.4.74
tan (+-e)=Me'/M'= -(TA4M')-1,
MNsin (+-e)=r-l
10.4.75 M"+xM- r - 2 ~ - 3 = ~
10.4.76 (M2)"'+4s(M2)'-2&f2=0
10.4.77 e'2+ q(e"'/e') -f ( e " / e y = X
Asymptotic Expansione of Modulus and Phase for
Large z
10.4.78 M2(z) 1 x-'"--7r
ao
0
ip-jg-23k
(-'Ik (i) I
(2~)-3k
10.4.79
-- (ZZ)-D+
82825
128 14336
10.4.80
10.4.68
10.4.81
Bi' (~e*"l~)
49527 1 2065 30429 (2s)-12+
+- 640 (2s) 2048 . . .]
Asymptotic Forms o f p (f t) & p for Large z
, f t)&
( i
(la% 4<
3
10.4.69
Modulus and Phase
10.4.82 I A i (t)dt-i-l 2 7r-1/2x-3/4
3 exp (-: 212)
Ai (--5)=M(z) cosB(z), Bi (-s)=M(s) sin e(r) 10.4.83
M(s)=d[Ai2 (--s)+Bi2 (-z)],
+)=arctan [Bi (-x)/Ai (-s)]
'See page 11.

16. 450 BESSEL FUNCTIONS OF FRACTIONAL ORDER
10.4.84 l (:
Bi ( t ) d t - ~ - ~ / ~ zexp / ~
-~ z3l2)
Asymptotic Forms of Gi (kz), (kz),Hi (kz), (kz)
Gi‘ Hi’
for Large z
10.4.86 Gi (z)-x-lz-’
10.4.87 Gi (-z) -7r-l?~-l/~ cos (; z3/2+3
-1080 56875 z-8
10.4.88 Gi’(z)
7
96
--
x-W2
69 67296
16 23755 96875 z-lo, . . .)
+ 3344 30208
sin
10.4.89 Gi’(--5)-1~-~/~2~/~ (f ill+$)
g(z)-z2/3(1-, 7 z-2+- 35 z-*-- 181223 z-6
- l / (3z312)
Hi (z)- ~ - ‘ / ~ z exp ~ 288 207360
10.4.90
10.4.91 Hi (-z) -7r-Iz-’ 186 83371 z-8
+ 12 44160
10.4.92 Hi‘(x) -x-1/2x1/4
exp (W2) -9 11458 84361
1911 02976
.. .)
10.4.93
Zeros and Their Asymptotic Expansions
Ai (z), Ai’(z) have zeros on the negative real
+23 97875 z-6-
6 63552
... )
axis only. Bi (z), Bi’(z) have zeros on the nega-
tive real axis and in the sector f<
xl arg z<7.
1r
$
a,, a:; b b: s-th (real) negative zero of Ai (z),
,
Ai’(2); Bi (z), Bi’(z), respectively. B,, /3:; 2 8 , -843 94709 z-6+
265 42080
.. .)
s-th complex zero of Bi (z), Bi’(z) in the sectors
&r<arg ,<
bz -+,x<arg z<-fx, respectively. Formal and Asymptotic Solutions of Ordinary Differ-
ential Equations of Second Order With Turning
10.4.94 a,= - j [ 3 ~ ( 4 ~ -1)/8] Points
10.4.95 a:= -g[3~(4~-3)/8] A n equation
10.4.96 Ai’(a,) = (- 1)8-!f1[3~(4s-l)/81 10.4.106 W”+a(z, X)W’+b(z, X)W=O
10.4.97 Ai (a:)=(-1)a-1g1[3a(4s-3)/81
in which X is a real or complex parameter and,
for fixed X, a(z, X) is analytic in z and b(z, A) is
10.4.98 b, = -f[ 3~ (49-3)/8] continuous in z in some region of the z-plane, may
be reduced by the transformation
10.4.99 b:= - g [ 3 ~ ( 4 ~ -1)/8]
10.4.107 W(z)=w(z) exp ( - i s ’ a ( t , h)dt)
10.4.100 Bi’(b,) = (- 1)8-!f1[3~(4s-3)/8]
10.4.101 Bi (b:) = (- 1)’g1[3r(4s- 1)/8] to the equation
10.4.102 @,=erf/3f (4s-l)+-
3i
4
In 2
1 10.4.108
w”+(o(z, X)w=O
10.4.103 @:=erf/3g (4s-3)+-
3i
4
In 2
1 q(z,
1 I d
X)=b(z, X)-- 4 uqz, A)-- 2 - a(z, ’1
dz
*See page 11.

17. BESSEL FUNCTIONS O FRACTIONAL ORDER
F 451
If p(z, X) can be written in the form 10.4.114
10.4.109 p(z, X)=X'p(z)+q(z, A) yo@)=Ai (-P x ) [1 + O(X-91
where q(z, X) is bounded in a region R of the z- =Bi (- X2%)[ 1 + O(X-91
yl(x) (1x1+m )
plane, then the zeros of p(z) in R are said to be For further representations and details, we refer
turning points of the equation 10.4.108. to (10.41.
The Special Case w"+[A*z+q(z, X)]w=O When z is complex (bounded or unbounded),
conditions under which the formal series 10.4.110
Let X=IX(eiU vary over a sectorial domain S: yields a uniform asymptotic expansion of a solu-
IXl>A,,(>O),o l S w S w 2 , and suppose that q(z, X) is tion are given in [10.121 if q(z, A) is independent
continuous in z for Ilr z< and X in S, and q(z, X) of X and IXI+m with fixed w, and in 110.141 if X
-2q.(z)~-. as
0
in S. lies in any region of the complex plane. Further
references are [10.2; 10.9; 10.101.
Formal Series Solution The General Case w"+[Agp(z)+q(z, A)]w=O
10.4.110
w(z)=u(z) 5 p.(z)X-*+X-'u'(z)
0
OD
J.,(z)X-" Let X=IXJefWwhere IXl2b(>O) and - - ? r l w ~ - ? r ;
suppose that p(z) is analytic in a region R and has
u" +Pzu=O a zero z=% in R, and that, for fixed A, q(z, A) is
analytic in z for z in R. The transformation
~(z)=c0, J.o(~)=~-f~l, co,e constants
, €=€(z),v=[p(z)/#"W(z), where [ is defined as
the (unique) solution of the equation
10.4.115
yields the special case
(n=O, 1,2, . . .) 10.4.116 -
+
dzv [X2€+j(€, X)]v=O, *
dE2
uniform hpnptotic Expand0118Of %lUtiOns
For z real, i.e. for the equation
10.4.111 9'' +[XSZ+ &, A) I
Y
' 0 Exumple:
where x varies in a bounded interval aSxSb that Consider the equation
includes the origin and where, for each fixed X in S, 10.4.117 y"+[Xz-((xz-~) s-~]Y=O
q(z, A) is continuous in x for a 5s5b, the following
asymptotic representations hold. for which the points x=O, are singular points
(i) If X is real and positive, there are solutions and x=1 is a turning point. It has the functions
yo(x), yl(x)such that, uniformly in x on a_<xlO, ZVA(AZ),A ( ~ z ) particular solutions (see
~Y as
9.1.49).
10.4.112 The equation 10.4.115 becomes
yo(s)=Ai ( - P x ) [1 +O(h-')] (A+- )
y,(x)=Bi ( - P Z ) [ ~ + O ( ~ - ~ ) ]
whence
and, uniformly in x on 0 5 z S b
10.4.113
yo@) =Ai (- W x )[1+ O(X-')] +Bi (- A%) O(X-l),
yl(z)=Bi (-X*flx)[l+O(X-l)]+Ai (-Xzfl~)o(X-~)
0-m)
Thus
(ii) If 52RXl0, A f O , there are solutions
YO(Z), y~(z)such that, uniformly in x on a<x<b, 10.4.118 v(€) =(Fy Y(Z)
'See page n.

19. BESSEL FUNCTIONS OF FRACTIONAL ORDER 453
Example 2. Compute &(x) for x=24.6. Modified Spherical Beseel Functiom
Interpolation in Table 10.3 yields for x=24.6
~ - ~ ~ e Z ” * ~ (x) = (-28) 3.934616
j21
To compute &~I,,++(x), &&L+,(x), n=O,
1, 2, . . . for values of x outside the range of
~ - ~ e Z ’ / ~ (2)= (-27)9.48683
j20
whence Table 10.8, use formulas 10.2.13, 10.2.14 together
with 10.2.4 and obtain values for the hyperbolic
j21(24.6)=.05604 29, &0(24.6)=.03896 98. and exponential functions from Tables 4.4 and
From the recurrence relation 10.1.19 there 4.15. In those cases when J$&I,,++(~) and
results MxI-,,-+(x) are nearly equal, i.e., when x is
j19(24.6)= .00890 67660 [.00890 701 sufficiently large, compute mxK,,+*(x) from
j18(24.6)=-.02484 93173 [-.02485 901 formula 10.2.15, for which the coefficients (n+ 3, k)
j1,(24.6)= -.04628 17554 [- .04628 161 are given in 10.1.9.
j1,(24.6)= -.04099 87086 [- .04099 881
Example 3. Compute J~*/xI,,~(x), J~*~xK,~~(x)
j16(24.6)=-.00871 65122 [-.(lo871 671 for z=16.2.
For comparison, the correct values,are shown in From 10.2.13, &&166/2(x) = (3 +2) sinh x/2-
brackets. 3 cosh 212‘; from Table 4.4, cosh 16.2=(6)5.4267
To compute j15(x)for x=24.6 by Miller’s de- 59950 and this equals the value of sinh 16.2 to the
vice, take, for example, N=39 and assume same number of significant figures. Hence
FdO=O, F,,=l. Using 10.1.19 with decreasing N,
i.e., FN-l=[(2N+l)/x]FN-FN+l,N=39, 38, . . ., 1/$~/16.216/2(16.2)(.06243 402371
=
1, 0, generate the sequence F38,
F3,, . . Fl, F,
, .,’
-.01143 118427)[(6)5.4267 599501
compute from Table 4.6, j0(24.6)=(sin 24.6)/24.6 = 338814.4594- 62034.29298
- -.02064 620296, and obtain the factor of pro- =276780.1664.
portionality To compute J4?r/16.2K6,,(16.2) use 10.2.17 and
p=jo(24.6)/Fo= .OOOOO 03839 17642. obtain
The value pF16 equals j16(24.6) to 8 decimals. 4 m 2 K 5 / 21( =re
6.2) 6
The final part of the computations is shown in
the following table, in which the correct values = (-7)2.8945 38069[.036932 604001
are given for comparison.
= (- 8) 1.0690 28283.
I
N j~ (24.6) To compute m , + 5 n ( , ) a
I, + 8xfor
3 I ,
value of z within the range of Table 10.9, obtain
15 -22704.71107 -. 00871 67391 -. 00871 674 from Table 10.9, & IQ ) .1&$12,D(x) for
&l &,
14 +78178.88236 +. 03001 42522 t. 03001 425
13 f114866.80811 +. 04409 93941 f. 04409 939 the desired value of x and use these as starting
12 + 47894. 44353 +. 01838 75218 t.01838 752 values in the recurrence relation 10.2.18 for
11 -66193.59317 -. 02541 28882 -. 02541 289
10 -109782. 76234 -. 04214 75392 -.04214 754 decreasing n.
9 -27523.39903 -. 01056 67185 -.01056 672 To compute mxK.++(z)for some integer n
8 +88524.85252 +. 03398 62526 f. 03398 625
7 i-88699. 11017 +. 03405 31532 f . 03405 315 outside the range of Table 10.9, obtain from
6 -34440.02929 -. 01322 21348 -.01322 213 10.2.15 or from Table 10.8, m x K + ( x ) ,
5 -106899. 12565 -. 04104 04602 -. 04104 046
4 - 13360.39272 -. 00512 92905 -.00512 929 WxKal2(x) for the desired value of x and use
3 +102011. 17704 +. 03916 38905 f. 03916 389
2 +42387. 96341 +. 01627 34870 f.01627 349 these as starting values in the recurrence relation
1 -93395. 73728 -. 03585 62712 -. 03585 627 10.2.18 for increasing n. If x lies within the
0 -53777.68747 -. 02064 62030 -. 02064 620
range of Table 10.9 and n>10, the recurrence
may be started with 43?rlzK19,2(x), -K21n(x)
It may be observed that the normalization of the obtained from Table 10.9.
sequence F , FN-l, . ., Focan also be obtained
N . Example 4. Compute r n K l l D ( x ) for x=3.6.
from formula 10.1.50 by computing the sum Obtain from Table 10.8 for x=3.6
u=g 0
(2k+l)F: and h d i n g p=l/& This
yields, in the case of the example, p=l/&=
.OOOOO 03839 177.

20. 454
The recurrence relation 10.2.18 yields successively
-J3~/3.6K5/2(3.6) -.01192 222
=
---.02461 718
-
-dm6K,/2(3.6) = -.02461 718
--.12072 034
-
&r/3.6KlI/2(3.6) .04942 4480
=
=.35122 533.
--
3.6
d m K r D ( 3 . 6 )= .01523 3952
= .04942.4480
+c6 718)
5
(.01523 3952)
(.02461
-- (.04942 4480)
3.6
+c6 034)
9
(.12072
As a check, the recurrencecan be carried out until
n=9 and the value of J m K l , n ( 3 . 6 ) so obtained
can be compared with the corresponding value
from Table 10.9.
To compute =I,,+&) when both n and z
~’(x),
{
BESSEL FUNCTIONS OF FRACTIONAL ORDER
respectively, the following formulas may be
used, in which d, d’
3!
u4
4!
V’
2!
2! 31-
denote approximations to e, c’
and u=y (d)/y (d), v= y ’(d’)/d’’y(d’).
’
c=d-~-2d -+2 --24d2
u3
+88d $-(88+720@)
+5856d28!
03
3!
-
U6
5!
-(105+76d‘3+24d’6)$
$
u“-(16640d+40320d4)~+. . .
c’=d’ 1-~---(3+2d’~)--(15+10d’~)-
-(945+756dt3+272dt6)$- . . .}
l-d -+- 3
Y’(C)=Y’(~) u’ u 3d2q?+14d-b
I 4
-(14+45@)g+471da fi
uQ u7
U
5!
-(1432d+l575d4)g+. . .
-(15d/3+14d’3$
US
-v4
4!
{
are outside the range o Table 10.9, use the device
f
described in t9.201.
Airy Functions
-(105d’3+101d’”+45d’4)~-. .}
.
To compute Ai(z), Bi(x) for values of x beyond Example 6. Compute the zero of y(z)=Ai(x)
1, use auxiliary functions from Table 10.11. -Bi(z) near d= -.4.
Example 5. Compute Ai(z) for 2=4.5. From Table 10.11,
Firet, for x=4.5,
[=#$”‘=6.363961029, c1=.15713 48403. -
g(-.4)=.02420 467, y‘( .4)= -.71276 627
Hence, from Table 10.11, f(-E) = .55848 24 and whence u=y(-.4)/y’(-.4)= -.03395 8776. From
thus the above formulas
Ai (4.5) ~=-.4+.03395 8776-.OOOOO 5221
=$(4.5)-u4(.5584824) exp (-6.36396 1029) +.ooooo 01 11 +.ooooo 0001
=$(.68658 905)(.55848 24)(.00172 25302) - .36604 6333.
_
Y’(c) (- .71276 627) 1 + .00023 0640
=
= .00033 02503. -.OOOOO 6527 -.OOOOO 0027 +.OOOOO0002)
To compute the zeros c, c’of a solution y(z) of = (- .71276 627)(1.00022 4088)
the equation y”-xy=O and of its derivative --.71292 599.
-

21. BESSEL FUNCTIONS OF FRACTIONAL ORDER 455
References
Texts Tables
[10.1] H . Bateman and R. C. Archibald, A guide to tables [10.17] H. K. Crowder and G. C. Francis, Tables of
of Bessel functions, Math. Tables Aids Comp. 1, spherical Bessel functions and ordinary Bessel
205-308 (1944), in particular, pp. 229-240. functions of order half and odd integer of the
[10.2] T. M. Cherry, Uniform asymptotic formulae for first and second kind, Ballistic Research Labora-
functions with transition points, Trans. Amer. tory Memorandum Report No. 1027, Aberdeen
Math. Soc. 68, 224-257 (1950). Proving Ground, Md. (1956).
[10.3] A. ErdBlyi et al., Higher transcendental functions, [10.18] A. T. Doodson, Bessel functions of half integral
vol. 1, 2 (McGraw-Hill Book Co., Inc., New order [Riccati-Bessel functions], British Assoc.
York, N.Y., 1953). Adv. Sci. Report, 87-102 (1914).
[10.4] A. ErdBlyi, Asymptotic expansions, California [10.19] A. T. Doodson, Riccati-Bessel functions, British
Institute of Technology, Dept. of Math., Assoc. Adv. Sci. Report, 97-107 (1916).
Technical Report No. 3, Pasadena, Calif. (1955). [10.20] A. T. Doodson, Riccati-Bessel functions, British
[10.5] A. ErdBlyi, Asymptotic solutions of differential Assoc. Adv. Sci. Report, 263-270 (1922).
equations with transition points or singularities, [10.21] Harvard University, Tables of the modified Hankel
J. Mathematical Physics 1, 16-26 (1960). functions of order one-third and of their deriva-
[10.6] H. Jeffreys, On certain approximate solutions of tives (Harvard Univ. Press, Cambridge, Mass.,
linear differential equations of the second order, 1945).
h c . London Math. SOC. 428-436 (1925).
23, [10.22] E. Jahnke and F. Emde, Tables of functions, 4th
[10.7] H. Jeffreys, The effect on Love waves of hetero- ed. (Dover Publications, Inc., New York, N.Y.,
geneity in the lower layer, Monthly Nat. Roy. 1945).
Astr. Soc., Geophys. Suppl. 2, 101-111 (1928).
[10.23] C. W. Jones, A short table for the Bessel functions
[10.8] H. Jeffreys, On the use of asymptotic approxima-
tions of Green's type when the coefficient has Zn++(z), (2/r)Kn+4(2) (Cambridge Univ. Press,
Cambridge, England, 1952).
zeros, Proc. Cambridge Philos. Soc. 52, 61-66
(1956). [10.24] J. C. P. Miller, The Airy integral, British h o c .
[10.9] R. E. Langer, On the asymptotic solutions of Adv. Sci. Mathematical Tables, Part-vol. B
differential equations with an application to the (Cambridge Univ. Press, Cambridge, England,
Bessel functions of large complex order, Trans. 1946).
Amer. Math. Soc. 34, 447-480 (1932). t10.251 National Bureau of Standards, Tables of spherical
[10.10] R. E. Langer, The asymptotic solutions of ordinary Bessel functions, vols. I, I1 (Columbia Univ.
linear differential equations of the second order, Press, New York, N.Y., 1947).
with special reference to a turning point, Trans. [10.261 National .Bureau of Standards, Tables of Bessel
Amer. Math. Soc. 67, 461-490 (1949). functions of fractional order, vols. I, I1 (Co-
[10.11] W. Magnus and F. Oberhettinger, Formeln und lumbia Univ. Press, New York, N.Y., 1948-49).
Siitze fur die speaiellen Funktionen der mathe- [10.27] National Bureau of Standards, Integrals of Airy
matischen Physik, 2d ed. (Springer-Verlag, functions, Applied Math. Series 52 (U.S. Gov-
Berlin, Germany, 1948). ernment Printing Office, Washington, D.C.,
[10.12] F. W. J. Olver, The asymptotic solution of linear 1958).
differential equations of the second order for [10.28] J. Proudman, A. T. Doodson and G. Kennedy,
large values of a parameter, Philos. Trans. Roy. Numerical results of the theory of the diffraction
Soc. London [A] 247, 307-327 (1954-55). of a plane electromagnetic wave by a conducting
[10.13] F. W. J. Olver, The asymptotic expansion of Bessel sphere, Philos. Trans. Roy. Soc. London [A]
functions of large order, Philos. Trans. Roy. 217, 279-314 (1916-18), in particular pp. 284-
Soc. London [A] 247, 328-368 (1954). 288.
[10.14] F. W. J. Olver, Uniform asymptotic expansions of [10.29] M. Rothman, The problem of an infinite plate
solutions of linear secondarder differential under an inclined loading, with tables of the
equations for farge values of a parameter, Philos. integrals of Ai (kz),Bi (+z), Quart. J. Mech.
Trans. Roy. Soc. London [A] 250,479-517 (1958). Appl. Math. 7, 1-7 (1954).
[10.15] W. R. Wasow, Turning point problems for systems [10.30] M. Rothman, Tables of the integrals and difEer-
of linear differential equations. Part I : The
ential coefficients of Gi (+z), Hi (--z), Quart. J.
formal theory; Part 11: The analytic theory.
Mech. Appl. Math. 7, 379-384 (1954).
Comm. Pure Appl. Math. 14, 657-673 (1961);
15, 173-187 (1962). [10.31] Royal Society Mathematical Tables, vol. 7,
[10.16] G. N. Watson, A treatise on the theory of Bessel Bessel functions, Part 111. Zeros and associated
functions, 2d ed. (Cambridge Univ. Press, values (Cambridge Univ. Press, Cambridge,
Cambridge, England, 1958). England, 1960).

22. 456 BESSEL FUNCl'IONS OF FRACTIONAL ORDER
[10.32] R. S. Scorer, Numerical evaluation of integrals of [10.33] A. D. Smirnov, Tables of Airy functions (and
the form special confluent hypergeometric functions).
Translated from the Russian by D. G. Fry
(Pergamon Press, New York, N.Y., 1960).
[10.34] I. M. Vinogadov and N. G. Cetaev, Tables of
Besael functions of imaginary argument (Izdat.
and the tabulation of the function Akad. Nauk SSSR., Moscow, U.S.S.R., 1950).
(10.351 P. M. Woodward, A. M. Woodward, R. Hensman,
Gi (2)= (l/r)Jmsin (uz+ l/3u8)du, H. H. Davies and N. Gamble, Four-figure tables
of the Airy functions in the complex plane,
Quart.J. Mech. Appl. Math. 3, 107-112 (1960). Phil. Mag. (7) 37, 236-261 (1946).