Bessel functionsoffractionalorder1
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Transcript of Bessel functionsoffractionalorder1
436 BEBBEL FUNCTIONS OF FRACTIONAL ORDEB
page
(01255) . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Table 10.8. Modified Spherical Beasel Functions-ordera 0, 1 and 2
&&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, 6s 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,
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
Ai(z) =&z-%-Y(-€); Bi(z)=z-l/rdf([) Ai'(z)= -42Y4e-(g(-€); Bi'(z)=z'/Wg(€)
10s . Table 10.11. Airy Functions ( 0521 -) . . . . . . . . . . . . . . 475
Auxiliary Functions for Large Positive Arguments
f( f I), g( f [) ; )=&I+/', c'= 1.5 (- .l) .5 (- .05)0, 6D Auxiliary Functions for Large Negative Arpen t a
Ai(+ =z-"*lfi(O COB €+jd€) sin €1 Bi(-z)=z-l~'ffa(€) €-fi(€) sin €1 Ai'(-~)=z'/~[g~([) sin €-g2(€) cos €1 Bi'(-z)=z'/Ygal€) sin €+gl(€) COB €1 f1(€) J f' (€1 9 91 (€) 9 92 (€1 ; €= j2"" p=.05( -.01)0, 6-7D
Table 10.12. Integrals of Airy Functions (OIz110). . . . . . . . . 478
Ai(t)dt, z=0(.1)7.5; Ai(--t)dt, z=O(.l)lO, 7D SD': s,' Bi(t)dt; 2=0(.1)2; Bi(--t)d, z=0(.1)10, 7D 1
Table 10.13. Zeros and Associated Valuea of Airy Functions and Their
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 5855) Modulus and Phaae of Bi'(@,), Bi(8:) 8=1(1)5, 3D
Derivatives (1 1 8 1 lo) . . . . . . . . . . . . . . . . . . . . 478
The author acknowledge8 the eseietance of Bertha H. Walter and Ruth Zucker in the preparation and checking of the tablea and graphs.
Ajay k Sharma
Ajay k Sharma
10. Bessel Functions of Fractional Order Mathematical Properties
10.1. Spherical Beeeel Functions
Definitions
Werential Equation 10.1.1 $0" +2m' +[ 2-n(n+ 1) ]w=O
(n=O,*l, f 2 , . . .) Particular solutions are the Spherical Bessel
functions of thefist kind
jm(z> =Wdn+i(z) the Spherical Bessel functions of the second kind
Ya(z) =AGYn++(z) ,
and the Spherical Bessel functions of the th&d kind
hP(z)=jn(z) +iya(z)=dGFHSt(z),
hia) (z> =jm(z)-iyn(z) = ~ ZH ; ~ + ( Z > . The pairs jn(z), ya(z) and hil)(z), hf)(z) are
linearly independent solutions for every n. For general properties sm the remarks after 9.1.1.
10.1.2 Ancending S e r i e a (&e 9.1.2.9.1.10)
2s 32' 'm(z)=l. 3 .5 . . . (2n+1) { '-1!(2n+3)
10.1.3 163.5. . . (2n-1) 3 3
Ym(z)=- .p+ 1 l! (1-24
' 2 V n ) (3z')' (3-2%) - . . .} (n=O, 1,2, . . .)
Reprementotiom by Elementary Functions
10.1.8
j,(z)=z-'[P(n+*, 2) sin (z-+W) +Q(n+h 4 COS (z-h l l
10.1.9 yn(Z)=(-l)m+lz-'[P(n+), Z) COS (Z+#TJT)
-Q(n+3, Z) sin (z+h)l
-9 (-l)&(n+$, 2k)(22)-* 0
="y (-l)&(n+*, 2k+1)(22)-"-' 0
(n=O, 1,2, . . .)
1680 16120 I 30240
437
Ajay k Sharma
Ajay k Sharma
-.3-
FIGUB~ 10.1. j&). n=0(1)3.
*See page II.
FIGUSID 10.3. js(~), va(Z). ~ = 1 0 .
Pohon's Integral and Gegenbauer's Generalization
10.1.13 jn(z)== 2' s, - cos (2 cos e) sin2"+'8 ds
=z 1 ( - i ) n l e'*coaep,,(cos e) sine&
(See 9.1.20.)
10.1.14
(n=O, 1,2, . . .)
BESSEL FUNCTIONS OF FEtACTIONAL ORDER 439
Spherical Beaeel Functions of the !bond and Third 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 *
0 hi2' (z) =iR+lz- le- iZ~ (n++ k) (2iz)
DHertmtiation Formulae
10.1.26
(See 9.2.28.)
10.1.28 (+X/Z)Mf,2 (2) =j:( 2) + y: (2) = 2-2
10.1.29
(+X/Z>M,~,~(Z) =j;(z) +y;(z) =~-~+2 - '
10.1.30
0 n-2k
(n=O, 1,2, . . .)
Analytic Continuation
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
*See Page n.
440 BESSEL FUNCTIONS OF FRACTIONAL ORDER
Some In6nite S e r i e e Involving j:(z)
10.1.50 2 (2n+l)j:(z)=l 0
sin 22 m
0 10.1.51 (-1)R(2n+1)j:(~)=3--
10.1.52
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,(z) are the same as the zeros of Jn+&) and Y,+)(z) and the formulas for 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
10.1.57
441 BESSEL FUNCTIONS OF FRACTIONAL ORDER,
McMahon's Expadons for n Fixed and a Large
10.1.58
d . 8 ~ b:,8-8-((c(+7) (8/3)-1
4 3 -- (7p2+ 154/~+95)(80))'~
-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+$)--"/3+ . . . 10.1.60
b:, - (n+ 3) + 1.821098O(n+ 3) 'p + .802728(n+ 4) -m-. 11740(n + 4)
+.0249(n+$)-6/3+ . . .
10.1.61
j,(a:, ') -.8458430(n+$) 4'" { 1 - .566032(n+3) -2/3 +.38081(n+3)[email protected](n+3)-a+ . . .}
10.1.62
&(b:, 1) 4183921 (n+ 4)-"'"{ 1-1 .274769(n+4))-2/3
+ 1.23038 (n+ 3) -4/3- 1.0070 (n+4) -2+ . . . } See [10.31] for corresponding expansions for
8=2, 3.
Uniform Asymptotic Expansions of Zeros and Aeeociated Values for n Large
10.1.63
11+~~xi(n+t)-"3b:i(n+3)-~} k= l
h&), z(€) are defined as in 9.5.26, 9.3.38, 9.3.39. a:, b; s-th (negative) real zero of Ai'(z), Bi'(z) (see 10.4.95, 10.4.99.)
Complex Zeros of h!"(z), h!"'(z)
hil)(z) and hi')(z.em**), m any integer, have the same zeros.
hi1) (z) has n zeros, symmetrically distributed with respect to the imaginary axis and lying approxi- mately on the finite arc joining z=-n and z=n shown in Figure 9.6. If n is odd, one zero lies on the imaginary axis.
hi*)'(z) has n + 1 zeros lying approximately on the same curve. If n is even, one zero lies on the imaginary axis.
BESSEL FUNCTIONS OF FRACTIONAL ORDER
-z ( -m i (r)
-. 4409724 -. 4572444 -. 4702250 -. 4802184 -. 4875705
-. 4926355
(-I)'& (z) . 00000 .00575 .01118 .01592 .01983
.02290
-. 122500 -. 114201 -. 10'1243 -. 101318 -. 096159
-. 091561
-. 06806 -. 05986 -. 05279 -. 04674 -. 04160
-. 03725
. 000000
.027518
.049069
.065677
.078255
.OS7587
0. 0 0. 2 0. 4 0. 6 0. 8
1. 0
.oooo
.0023
.0043
.0061
.0075
.0085
--3 hi (r) Hi (r)
-. 09156 -. 05056 -. 03043 -. 01950 -. 01310
-. 00914 -. 00658 -. 00485 -. 00366 -. 00280
-. 00219 -. 00173 -. 00138 -. 00112 -. 00091
-. 00075 -. 00062 -. 00052 -. 00044 -. 00037
-. 00032 -. 00027 -. 00023 -. 00020 -. 00018
-. 00015 -. 00013 -. 00012 -. 00010 -. 00009
-. 00008
-. 037 -. 014 -. 006 -. 003 -. 001
.OS7587
.065507
.050524
.039890
.032085
.026206
.021682
.018141
.015326
.013061
.011217
.009701
.008443
.007391
.006505
.005753
.005111
.004560
.004085
.003672
.003313
.002998
.002722
.002478
.002262
.002070
.001899
.001746
.001609
.001486
.001375
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 6. 2 5. 4 5. 6 5. 8
6. 0 6. 2 6. 4 6. 6 6. 8
7. 0
.0229
.0121
.0070
.0042
.0027
.0018
.0012
.OW8
.0006
.0004
.0003
.0002
.0002
. 0001
.om1
-. 4926355 -. 4131280 -. 3551700 -. 3108548 -. 2757704
-. 2472521 -. 2235898 -. 2036314 -. 1865701 -. 1718217
-. 1589519 -. 1476304 -. 1376005 -. 1286601 -. 1206469
-. 1134296 -. 1069004 -. 1009699 -. 0955634 -. 0906180
-. 0860804 -. 0819049 - .0780523 -. 0744888 -. 0711850
-. 0681152 -. 0652570 - .0625905 -. 0600985 -. 0577653
-. 0555773
0. 40 .36 .32 .28 .24
-. 0645731 -. 0487592 -. 0352949 -. 0242415 -. 0155683
-. 0091416 -. 0047276
-. 00013 --. 00005 -. 00002 -. 00001
.001859
.001056
.OW551
.000259
.000106
.20
. 16
. 12
.08
.04
. oooo37
.000010
.000002 . - -. - .
-. 0020068 -. 0005965 -. 0000747
I . oo I -.0000000 I
443 BESSEL F"I0NS OF FRAcTlONAL ORDER
10.2. Modified Spherical Bessel Functions
Definitions
Differential Equation 10.2.1
z2w" +2 zw' -[ z2+n(n+ 1) ]w = 0 (n=O, f l , f2, . . .)
Particular solutions are the Modifid Spherka? Bessel fum!;.nS of the$rst kind,
10.2.2
&$&++(z) = e-nri/*jn(zer*/a) (-r<arg z 5 %.> = e3nrf/2jn(~e-3rt/a) (I)lr<arg z 27)
of the second kind,
10.2.3
-I- ,++ (z) = ea(n+l)rt/ (2 e (-r<arg 2 5 31r)
(h<arg z5r) -e- - (n+1)r t/%~(~~ -3r t/a )
of the third kind,
10.2.4
r n K + + ( Z ) = fr(-l)"+'dm[In+t(z) -I--)I-+(Z)l
WzIn++(z), r n Z I - n - + ( Z )
d G m n + + ( Z ) 9 r nZKn++(Z >
The pairs
and
are linearly independent solutions for every n. Most properties of the Modified Spherical
Bessel functions can be derived from those of the Spherical Bessel functions by use of the above relations.
Ascending %rim 10.2.5
10.2.6 1 . 3 . 5 . . . (2n-1)
4G I - n - + ( Z ) = (-1)"2"+1
3za + { '+1!(1-2n) 2!(1-%)(3-%)
(n=O, 1,2, . . .) f
(n=O, 1,2, . . .) (See 10.1.9.)
10.2.12
&&zI,++(z)=g,(z) sinh z+g-n-l(z) cosh z go( 2) = 2-1, g1(z) = - 2-2
gn-1(4 -gn+1(z) = (2n+ 1) z-'gn(z> (n=O, f l , f2, . . .)
The Functions 1/&&2I*cn++,(Z), n=O, 1, 2 .
10.2.13 sinh z
r n I l , 2 ( Z > =2
sinh 2-- 3 cosh z Z2
10.2.14 cosh z
MZ I - l / 2 ( Z ) =- Z
sinh z cosh z Z 22
&+I -3,2 (2) =-- -
3 22
&&I-&)=-- sinh z+
*See page 11.
444 BESSEL FUNCTIONS OF
Modified Spherical Besee1 Functions of the Third Kind
10.2.15
&&zKn++(z) =3?rie"+1)~"2h~~)(ze*'f) (-*<wg z 53.1
O,,.<arg 2 5 4 --3,,.ie- - (n+1)ri/2h(2) (ze-+rt)
I
=($?r/z)e-Z 5 (n++, k)(22)-~ 0
10.2.16
K,+*(z)=K-~-~(z) (n=O, 1 ,2 , . . .)
The Functions mRn++(~ ) , n=O, 1, 2
10.2.17 &@&(z)= (h/z)e-'
&?2GI2(z) = ($n/z)e-Yl+z-')
mK~,2(Z)=(3?r/Z)e-'(l+32-'+32-e)
Elementary Properties
Recurrence Relations
f&): r n L + * ( Z ) , (-l)"+'mKn++(Z)
(n=O, f l , &2, . . .)
10.2.18 fn-l(z) -fn+1(2) = (2n4-1) z-'fn(z)
10.2.19 nfn-l(z)+(n+l)fn+l (z>=@n+1) s fn(z> d
n+l d 10.2.20 7 fn(z) +& fn(4 =fn-1(z)
10.2.21 -- fn(z) +-&fn(z) =fn+1(z)
(See 10.2.22.)
n d Z
(See 10.2.23.)
DifYerentiation Formulae
fn(2): r n L + * ( Z ) , (-w+1m&++(4 (n=O, kl, ~ 2 , . . . )
10.2.22 (; g J k+ 'f.(z) I= Zn-m+'f,-,(z)
10.2.23 (5 -g>l [z - y n (2) I = z - n-mfn+m ( 2)
(m=1,2,3,. . . )
FRACTIONAL ORDER
FIQURE 10.5. -&&+*(z), &K,.++(z). z=10.
BESSEL FUNCTION8 OF FRACTIONAL ORDER 4.45 Formulas of Rayleigh's Type
10.2.25
(n=O, 1,2, . . . )
(22)%-*" =- l n (.-1)k+1 (2n-k) ! (2n-2k)! 2 3 0 k! [ (n-k) I]*
Generating Functiom
16.2.30
(2 I t I < I z I) 10.2.31
Derivativea With Reapect to Order
10.2.32
10.2.33
For El(z) and Ei(z), see 5.1.1, 5.1.2.
*See page n.
Addition Theorems and Degenerate Forms
T, p, 6, X arbitrary complex; R=,@+p2-2rp coa 6
10.2.35
10.3. Riccati-Beasel Functions
DilTerential Equation
10.3.1
9w"+ [z5 -n(n + l)]w= 0 (n=O, f l , f 2 , . . .)
zjn(z), q/n(z) &p' (z), zhp (2)
Pairs of linearly independent solutions are
All properties of these functions follow directly from those of the Spherical Besael functions.
The Functiom zj,(z) zy&) n=O, 1, 2
10.3.2
zy'o(z)=sin 2, Zj,(z)=z-'sin 2-cos 2 zy'2(Z)=(3Z-2-1) S h 2-32' COB 2
10.3.3 q/&)=-cos 2, q/1(2)=-sin 2-2-1 cos 2
2&(2)=-32-' Sin 2-(32-'-1) COB 2
WTOMkiIUM
10.3.4 W{zy'n(z), zYn(z)}=l
10.3.5 W{ z@)(z), zhF)(~)} =-2i (n=O, 1,2, . . .)
446 BESSEL FUNCTIONS
10.4. Airy Functions
Definitions and Elementary Properties
Differential Equation
10.4.1 w" -zw=o Pairs of linearly independent solutions are
Ai (z), Bi (z), Ai (z) , Ai (2W3) , Ai (z), Ai ( ~ e - ~ ~ ~ / ~ ) .
Ascending Series
10.4.2 Ai (z) =clf(z) -c2g(z)
10.4.3 Bi (z) = & [cJ(z) +c~g(z)l 1 1-4 1.4.7 f(z)=i+- z3+- z6+- zO+ . . . 3! 6! 9!
=$ 3k (;)k G! 2 2.5 2.5.8
g(z)=z+- z4+- z7+- zl0+ . . . 4! 7 ! lo! 23k+1
=$ 3k (g)k (3k+l)!
(a+$),=l
3k (a+i)k=(3sl+1)(3a+4) . . . (&+3k-2) (a arbitrary; k=l, 2, 3, . . .)
(See 6.1.22.)
10.4.4
el=& (O)=Bi (0)/8=3-~fi/r(2/3)
10.4.5 c2=-Ai' (@)=Sif (0)/fi=3-ifl/r(1/3)
=.35502 80538 87817
=.25881 94037 92807
Relations Between Solutions
10.4.6- Bi (2) =erfn Ai (a"'/"> +e+@ Ai (ze-*"fl)
10.4.7 Ai (z)+e"fn Ai (a2rtD) +e-2*iD Ai (a-2rfn) =o
10.4.8 Bi (2) +e*fn Bi (=2r*/g> +e--2rfn Bi (=-*rim) =O
10.4.9 Ai (~e*~~'/~)=)e*'~/~[Ai (2) G Bi (z)]
Wl-OMkifUl9
W{Ai (z), Bi (z)} =T - ~ 10.4.10
10.4.11 W{Ai (z), Ai ( ~ e ~ * ~ / ~ ) } =+r-1e-rf/6 I
OF FRACTIONAL ORDER
10.4.12 W{Ai (z), Ai ( ~ e - ~ * ~ / ~ ) } =&r-lerfl6
10.4.13 W{ Ai ( ~ 6 ~ ~ ~ 1 ~ ) , Ai ( ~ e - ~ * ~ / ~ ) } =4ir-1
.." ~
FIGURE 10.7. Bi (~zz), Bi' (*4.
BESSEL FUNCTIONS OF FRACTIONAL ORDER 447
Representations of Beosel Functions in Terms of Airy Functions
10.4.22 5*1/3({)=)m[& Ai (-z)FBi (-2))
90.4.23 H:'l,3({)=e~d/6m[Ai (-z)-i Bi (-z)]
10.4.24 H2',,,({)=efd1"m[Ai (-z)+i Bi (-z)]
10.4.25 1*l/a({)=$mz[r & Ai (z)+Bi (z)]
10.4.26 K*ln(l) =rm Ai (z) 10.4.27 J,tzn({)=(J3/2z)[fJ3 Ai' (-z)+Bi'(-z)]
10.4.28 Hiti ({) = e-2rf /3H(l)
-21&)
=erf~'(J3/z)[Ai' (-2)-i Bi' (-41 10.4.29
Hi~~({)=e2rf/3H~z~1a({) (J3/z)[Ai' (- z) +i Bi' (- z)]
'See page n.
10.4.30 I*2n({)=(@/2z)[f43 Ai'(z)+Bi'(z)]
10.4.31 K*2n({) = -T ( ~ / z ) Ai' (z)
Integral Representations
(3a)-% Ai [ f (3a)-lBz]=Lm cos (at3ffzt)dt
10.4.32
10.4.33
(3a)-lB7r Bi [ f (3a)-lnz] =l [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=- [J-113(t)+J113(t)]dl :sd
Ascending Series for is Ai ( f Qdt, is Bi ( f t)dt 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)
10.4.40 1 Bi (t)dt=@[clF(z)+c2G(z)l
(See 10.4.3.) 10.4.41
s,' Bi (- t)dt= - fi[clF(- z) +c2G(- z) I 1 1.4 1.4.7 F(z)=z+- z4+- z7+- ,lo+ . . . 4! 7! 1 O!
1 2 2 . 5 2 . 5 . 8 G(z)=B z'+- z~+-z'+-z~'+ . .
p + 2
5! 8! ll!
=$ " (:)k (3k+2)! The constanta cy, c2 are given in 10.4.4, 10.4.5.
448 BESSEL FUNCI’IONS OF FRACTIONAL ORDER
Tbe Functions Gi(z), Hi+) 10.4.42
Gi ( ~ )=r - ~ ~ ~ s i n ( i t ~+z t ) d t
=IBi (z)+ ’[Ai (z) Bi (t)-Ai (t) Bi (z)]dt 3 s,
10.4.43
Gi’ @)=:Bit (z)+J’[Ai’(z) Bi (t)-Ai (t) Bi’(z)]dt
10.4.44
3 0
Hi(z)=s-’fom erp(-i P+zt) dt
=-Bi (z)+ [Ai (t) Bi (2)-Ai(z) Bi(t)]dt 3 “ s , ’
s,’ 10.4.45
2 Hi’(z)=3Bi’ (z)+
10.4.% Gi (z)+Hi (z)=Bi (z)
[Ai (t) Bi’(z)-Ai’(z) Bi (t)]dt
Representations of 1’ Ai( f t)&, s’ Bi( f t)& by Gi (kz), Hi (*z)
1 b i ( t) d t = -+ T[ Ai’ (z) Gi (z) -Ai (z) Gi ’ (z) ] 3
10.4.47
10.4.48
=--- n[Ai’ (2) Hi (2)-Ai (z) Hi’ (z)] 3
10.4.49
b i (-t)dt=----?r[Ai’ (-2) Gi (-2) 1 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’ (2) Gi (2)-Bi (2) Gi’ (z)]
10.4.52
10.4.53
l* Bi (-t)dt=--?r[Bi’ (-2) Gi (-2)
10.4.54 =*[Bi’ (-2) Hi (-2)
=-r[Bi’ (z) Hi (2)-Bi (z) Hi’(z)]
-Bi (-2) Gi’ (-z)]
-Bi (-2) Hi’ (-z)]
Difterential Equations for Gi (z), Hi (z)
10.4.55 w’ -zw=-?r-l
1 1 w(0) =- Bi (0) =- Ai (0) = .20497 55424 78 3 6
3 6 1 w’(O)=-Bi’ ( O ) = - l Ai’ (0)=.14942 94524 49
w(z)=Gi(z) w’ ’ - zw= r- 1 10.4.56 -
2 w(O)=- Bi (O)=A Ai (0)=.40995 10849 56 3 6
3 fl 2 2 w’(O)=- Bi’ (O)=-- Ai’ (0)=.29885 89048 98
w(z)=Hi (z)
Differential Equation for Products of Airy Functions
10.4.57 w’ “ -4ZW’ -2w=o
Linearly independent solutions are Ai2 (2)) Ai (z) Bi (z), Bi2 (2).
Wronskian for Products of Airy Functions
10.4.58 W{Ai2 (2)) Ai (z) Bi (2)) Bi2 (z) ]=27r3
Asymptotic Expansions for Iz( Large
ck (k=1,2, 3, . . .) 6k+l 6k-1
+I a=-- ) k
BESSEL FU" IONS OF FRACTIONAL ORDER 449 10.4.62 Ai' (-Z)--T-+Z*
10.4.68
Bi' (~e*"l~)
(la% 4<3 Modulus and Phase
10.4.69 Ai (--5)=M(z) cosB(z), Bi (-s)=M(s) sin e(r)
M(s)=d[Ai2 (--s)+Bi2 (-z)], +)=arctan [Bi (-x)/Ai (-s)]
10.4.70
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
- r-l, W+'= - *-Ix 10.4.71
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-'" ao ip-jg- (-'Ik 23k (i) (2~)-3k 0 I 7r
10.4.79
-- 82825 (ZZ)-D+ 14336 128
10.4.80
10.4.81
. . .] 49527 1 640
2065 30429 (2s) -12+ +- (2s) 2048
Asymptotic Forms o f p ( f t) &,pi ( f t)& for Large z
10.4.82 I A i (t)dt-i-l 3 2 7r-1/2x-3/4 exp (-: 2 1 2 ) 10.4.83
'See page 11.
450 BESSEL FUNCTIONS OF FRACTIONAL ORDER
10.4.84 l Bi ( t ) d t - ~ - ~ /~ z - ~ /~ exp (: z3l2)
Asymptotic Forms of Gi (kz), Gi‘ (kz), Hi (kz), Hi’ (kz) for Large z
10.4.86 Gi (z) -x-lz-’
10.4.87 Gi (-z) -7r-l?~-l/~ cos (; z 3 / 2 + 3
7 96 Gi’ (z) -- x-W2 10.4.88
10.4.89 Gi’ (--5)-1~-~/~2~/~ sin (f ill+$) 10.4.90
10.4.91 Hi (-z) -7r-Iz-’
10.4.92
Hi (z) -~ -‘/~z- l/~ exp ( 3z312)
Hi‘ (x) -x-1/2x1/4 exp (W2)
10.4.93
Zeros and Their Asymptotic Expansions
Ai (z), Ai’ (z) have zeros on the negative real axis only. Bi (z), Bi’ (z) have zeros on the nega- tive real axis and in the sector fx<l arg z1<$7r. a,, a:; b,, b: s-th (real) negative zero of Ai (z), Ai’ (2); Bi (z), Bi’ (z), respectively. B,, /3:; 2 8 , s-th complex zero of Bi (z), Bi’ (z) in the sectors &r<arg z<b, -+,x<arg z<-fx, respectively.
10.4.94 a,= - j [ 3~ (4~ - 1)/8]
10.4.95 a:= -g[3~(4~-3)/8]
10.4.96 Ai’ (a,) = (- 1)8-!f1[3~(4s- l)/81
10.4.97 Ai (a:)= (-1)a-1g1[3a(4s-3)/81
10.4.98 b, = -f[ 3~ (49 - 3) /8]
10.4.99 b:= -g [3~(4~- 1)/8]
10.4.100 Bi’ (b,) = (- 1)8-!f1[3~(4s-3)/8]
10.4.101 Bi (b:) = (- 1)’g1[3r(4s- 1)/8]
10.4.102 @,=erf/3f (4s-l)+- In 2 3i 4 1 10.4.103 @:=erf/3g (4s-3)+- In 2 3i 4 1
1080 56875 z-8 - 69 67296
16 23755 96875 z-lo, . . .) + 3344 30208
g(z)-z2/3(1-, 7 z-2+- 35 z-*-- 181223 z-6
288 207360
186 83371 z-8
+ 12 44160 - 9 11458 84361
. . .) 1911 02976
+23 97875 z-6- . . . ) 6 63552
. . .) 843 94709 z-6+ - 265 42080
Formal and Asymptotic Solutions of Ordinary Differ- ential Equations of Second Order With Turning Points
An equation
10.4.106 W”+a(z, X)W’+b(z, X)W=O
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 continuous in z in some region of the z-plane, may be reduced by the transformation
10.4.107 W(z)=w(z) exp (- i s’a(t , h)dt)
to the equation
10.4.108 w”+(o(z, X)w=O
1 I d q(z, X)=b(z, X)-- 4 uqz, A)-- 2 - dz a(z, ’1
*See page 11.
BESSEL FUNCTIONS OF FRACTIONAL ORDER 451
If p(z, X) can be written in the form
10.4.109 p(z, X)=X'p(z)+q(z, A)
where q(z, X) is bounded in a region R of the z- plane, then the zeros of p(z) in R are said to be turning points of the equation 10.4.108.
The Special Case w"+[A*z+q(z, X)]w=O
Let X=IX(eiU vary over a sectorial domain S: IXl>A,,(>O), o lSwSw2 , and suppose that q(z, X) is continuous in z for Izl<r and X in S, and q(z, X)
-2 q.(z)~-. as in S.
10.4.110
0
Formal Series Solution
OD
w(z)=u(z) 5 p.(z)X-*+X-'u'(z) J.,(z)X-" 0
u" +Pzu=O
~(z)=c0, J.o(~)=~-f~l, co, e, constants
(n=O, 1,2, . . .)
uniform hpnptotic Expand0118 Of %lUtiOns
For z real, i.e. for the equation
10.4.111
where x varies in a bounded interval aSxSb that includes the origin and where, for each fixed X in S, q(z, A) is continuous in x for a 5s 5 b, the following asymptotic representations hold.
(i) If X is real and positive, there are solutions yo(x), yl(x) such that, uniformly in x on a_<xlO,
10.4.112
9'' + [ XSZ+ &, A) IY' 0
yo(s) =Ai ( - P x ) [ 1 + O(h-')] y,(x)=Bi ( -PZ)[~+O(~-~) ]
(A+- )
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 (-Xzf l~)o(X-~)
0 -m )
(ii) If 52RXl0, A f O , there are solutions YO(Z), y~(z) such that, uniformly in x on a<x<b,
'See page n.
10.4.114 yo@) =Ai ( - P x ) [ 1 + O( X-91 yl(x) =Bi (- X2%)[ 1 + O(X-91 (1x1 +m )
For further representations and details, we refer to (10.41.
When z is complex (bounded or unbounded), conditions under which the formal series 10.4.110 yields a uniform asymptotic expansion of a solu- tion are given in [10.121 if q(z, A) is independent of X and IXI+m with fixed w, and in 110.141 if X lies in any region of the complex plane. Further references are [10.2; 10.9; 10.101.
The General Case w"+[Agp(z)+q(z, A)]w=O
Let X=IXJefW where I X l 2 b ( >O) and - - ? r lw~ - ? r ; suppose that p(z) is analytic in a region R and has a zero z=% in R, and that, for fixed A, q(z, A) is analytic in z for z in R. The transformation €=€(z), v=[p(z)/#"W(z), where [ is defined as the (unique) solution of the equation
10.4.115
yields the special case
10.4.116 dzv -+ [X2€+j(€, X)]v=O, * dE2
Exumple:
Consider the equation
10.4.117 y"+[Xz-((xz-~) s-~]Y=O
for which the points x=O, are singular points and x=1 is a turning point. It has the functions ZVA(AZ), ~ Y A ( ~ z ) as particular solutions (see 9.1.49).
The equation 10.4.115 becomes
whence
Thus
10.4.118 v(€) =(Fy Y(Z)
BESSEL FUNCTIONS OF FRACTIONAL ORDER 453
-22704.71107 +78178.88236
f114866.80811 + 47894. 44353 -66193.59317 - 109782. 76234 -27523.39903 +88524.85252 i-88699. 11017 -34440.02929
-106899. 12565 - 13360.39272
+102011. 17704 +42387. 96341 -93395. 73728 -53777.68747
Example 2. Interpolation in Table 10.3 yields for x=24.6
Compute &(x) for x=24.6.
~ - ~ ~ e Z ” * ~ j 2 1 (x) = (-28) 3.9346 16 ~ - ~ e Z ’ / ~ j 2 0 (2) = ( - 27) 9.48683
whence j21(24.6)=.05604 29, &0(24.6)=.03896 98.
From the recurrence relation 10.1.19 there results
j19(24.6)= .00890 67660 [.00890 701 j18(24.6)=-.02484 93173 [-.02485 901 j1,(24.6) = - .04628 17554 [- .04628 161 j1,(24.6) = - .04099 87086 [- .04099 881 j16(24.6)=-.00871 65122 [-.(lo871 671
For comparison, the correct values,are shown in brackets.
To compute j15(x) for x=24.6 by Miller’s de- vice, take, for example, N=39 and assume 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, 0, generate the sequence F38, F3,, . . . ,’ Fl, F,, compute from Table 4.6, j0(24.6)= (sin 24.6)/24.6 - - - .02064 620296, and obtain the factor of pro- portionality
p=jo(24.6)/Fo= .OOOOO 03839 17642.
The value pF16 equals j16(24.6) to 8 decimals. The final part of the computations is shown in the following table, in which the correct values are given for comparison.
-. 00871 67391 +. 03001 42522 +. 04409 93941 +. 01838 75218 -. 02541 28882 -. 04214 75392 -. 01056 67185 +. 03398 62526 +. 03405 31532 -. 01322 21348 -. 04104 04602 -. 00512 92905 +. 03916 38905 +. 01627 34870 -. 03585 62712 -. 02064 62030
N
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
I
j~ (24.6)
-. 00871 674 t. 03001 425 f. 04409 939 t. 01838 752 -. 02541 289 -.04214 754 -.01056 672 f. 03398 625 f. 03405 315 -.01322 213 -. 04104 046 -.00512 929 f. 03916 389 f.01627 349 -. 03585 627 -. 02064 620
It may be observed that the normalization of the sequence FN, FN-l, . . ., Fo can also be obtained from formula 10.1.50 by computing the sum u = g (2k+l)F: and hding p=l/& This
yields, in the case of the example, p=l/&= .OOOOO 03839 177.
0
Modified Spherical Beseel Functiom
To compute &~I,,++(x), &&L+,(x), n=O, 1, 2, . . . for values of x outside the range of Table 10.8, use formulas 10.2.13, 10.2.14 together with 10.2.4 and obtain values for the hyperbolic and exponential functions from Tables 4.4 and 4.15. In those cases when J$&I,,++(~) and MxI-,,-+(x) are nearly equal, i.e., when x is sufficiently large, compute mxK,,+*(x) from formula 10.2.15, for which the coefficients (n+ 3, k) are given in 10.1.9.
Example 3. Compute J~*/xI,,~(x), J~*~xK,~~(x) for z=16.2.
From 10.2.13, &&166/2(x) = (3 +2) sinh x/2- 3 cosh 212‘; from Table 4.4, cosh 16.2=(6)5.4267 59950 and this equals the value of sinh 16.2 to the same number of significant figures.
1/$~/16.216/2(16.2) = (.06243 402371
Hence
-.01143 118427)[(6)5.4267 599501 = 338814.4594- 62034.29298 =276780.1664.
To compute J4?r/16.2K6,,( 16.2) use 10.2.17 and obtain
4 m 2 K 5 / 2 ( 1 6.2) =re 6
= (-7)2.8945 38069[.036932 604001
= (- 8) 1.0690 28283.
To compute mI,,++(x), 35n I 8 , for a value of z within the range of Table 10.9, obtain from Table 10.9, &&IlQ&), .\1&$12,D(x) for the desired value of x and use these as starting values in the recurrence relation 10.2.18 for decreasing n.
To compute mxK.++(z) for some integer n outside the range of Table 10.9, obtain from 10.2.15 or from Table 10.8, mxK+(x), WxKal2(x) for the desired value of x and use these as starting values in the recurrence relation 10.2.18 for increasing n. If x lies within the range of Table 10.9 and n>10, the recurrence may be started with 43?rlzK19,2(x), -K21n(x) obtained from Table 10.9.
Compute rnK l lD (x ) for x=3.6. Example 4. Obtain from Table 10.8 for x=3.6
454 BESSEL FUNCTIONS OF FRACTIONAL ORDER
The recurrence relation 10.2.18 yields successively
- J3~/3.6K5/2(3.6) = - .01192 222
-- (.01523 3952) 3.6
- ---.02461 718
dmKrD(3 . 6 ) = .01523 3952 5 +c6 (.02461 718)
= .04942.4480
-dm6K,/2(3.6) = - .02461 718
-- (.04942 4480) 3.6
-- - .12072 034
&r/3.6KlI/2(3.6) = .04942 4480 9 +c6 (.12072 034)
=.35122 533.
As a check, the recurrence can be carried out until n=9 and the value of JmKl ,n (3 .6 ) so obtained can be compared with the corresponding value from Table 10.9.
To compute =I,,+&) when both n and z are outside the range of Table 10.9, use the device described in t9.201.
Airy Functions
To compute Ai(z), Bi(x) for values of x beyond
Example 5. Compute Ai(z) for 2=4.5. Firet, for x=4.5,
1, use auxiliary functions from Table 10.11.
[=#$”‘=6.36396 1029, c1=.15713 48403.
Hence, from Table 10.11, f( -E) = .55848 24 and thus Ai (4.5)
=$(4.5)-u4(.55848 24) exp (-6.36396 1029) =$(.68658 905)(.55848 24)(.00172 25302) = .00033 02503.
To compute the zeros c, c’ of a solution y(z) of the equation y”-xy=O and of its derivative
~’(x) , respectively, the following formulas may be used, in which d, d’ denote approximations to e, c’ and u = y (d)/y ’ (d) , v= y ’ (d’)/d’’y(d’).
c=d-~-2d -+2 u3 --24d2 u4 U 6 -
3! 4! 5!
+88d $-(88+720@) $ +5856d2 u“-(16640d+40320d4)~+ 8! . . .
c’=d’ 1-~---(3+2d’~)--(15+10d’~)- V’ 0 3 -v4
3! 4! { 2!
- (105+76d‘3+24d’6)$
-(945+756dt3+272dt6)$- . . .}
I 4 U b Y’(C)=Y’(~) l-d u’ -+- u3 3d2q?+14d- { 2! 31- 5! -(14+45@)g+471da uQ fi u7
-(1432d+l575d4)g+. US . .
- (15d/3+14d’3$
-(105d’3+101d’”+45d’4)~- . . .}
Compute the zero of y(z)=Ai(x) Example 6. -Bi(z) near d= -.4.
From Table 10.11,
g( - .4) = .02420 467, y‘( - .4) = - .71276 627
whence u=y(-.4)/y’(-.4)= -.03395 8776. From the above formulas
~=-.4+.03395 8776-.OOOOO 5221 + .ooooo 01 11 + .ooooo 0001
-_ - .36604 6333. Y’(c) = (- .71276 627) { 1 + .00023 0640
- .OOOOO 6527 - .OOOOO 0027 + .OOOOO 0002) = (- .71276 627) (1.00022 4088) -- - .71292 599.
BESSEL FUNCTIONS OF FRACTIONAL ORDER
References
455
Texts
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456 BESSEL FUNCl'IONS OF FRACTIONAL ORDER
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