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Unformatted text preview: EORY 0F BESSEL F . =
UNCTIONS [CHAP ‘ _] FUNCTIONS OF LARGE ORDER 249 e extent to which the condi tion iar Z < ’ h"
0“ fOr‘mﬂae <5>—<8>, is rem g l i” ‘V v. bl i: the last expression is one of Airy’s integrals (§'6'4). It follows that,
ova e. th ' t ' I i“ n,
e e in egrand in (2), qua function of 'r, are the valui . 1 2(n—x)§ 29(n—w)§
) fails to be a monogenic function of 'T, so that ~31 J7l(m)~7—ri 35*} Ki 3.70% __> ’
198 Of T correspondmg to those values of w for which; i when w > n,
dr/dw = 0. ‘ ' _ 1 2 (as — n) t
oints V (2) {J_§+Jil,
'7' = 27nd, l 1 en the arguments of the Bessel functions on the right are % {2 (m — n)}9/x*.
gra va ues. The corresponding formula for Y,,(ac) when as> n was also found by
.‘ holson; with the notation employed in this work it is 2 (x — n)
,(3) new—{e 3,,
3, The chief disadvantage of these formulae is that it seems impossible to : ' mine, by rigorous methods, their domains of validity and the order of
' 'tude of the errors introduced in using them. i g the contour through any angle i
3V(eithler positively or negatively), and weitheli
tinuatlon of H,‘”(2) or Hyl2l(z) over the range
1;. By givnig 7; suitable values, we thus ﬁnd th 5
re valid over the extended reg A ~7r<argz<m * l
} .{J_, —J,,. ion to real 'variables, we see that ietermined asymptotic expansions of J, (x) valid when x ‘ I
x/u_'> 1, (iii) {x~ Vi not large compared with xi But t i
5en (1) and (iii) and also between (ii) and (iii) and in H: L
,rly'equal to 1 while {Jr—pl is large. In these transitio
olvmg elementary functions only in each term) do not exi :
irmulae have been discovered by Nicholson which invol:
formulae of this type will now be investigated. the solutionof the problem With a View to remedying this defect, Watson“ examined Debye’s ‘grals, and discovered a method which is theoretically simple (though
in: it is very laborious), by means of which formulae analogous to
[wholson’s are obtained together with an upper limit for the errors involved. ; The method employed is the following: Debye’s integral for a Bessel function Whose order 12 exceeds its argument
(Evsech a) may be written in the form[ I 4 . . . . V t ha“ :0 "i mlulae valtd m the transitional regzons. , J” (V sech 0‘) = if + e” dw’ uaeof '—' ' " ' i W 60—1". sec d 8 4 ~ 8 4»? m the transxtional regions led: 7 9,9 7 = — sinh a (cosh w — 1) — COSh 0‘ (Sinh w _ w)
on approx1matlons to Bessel’s integral ingthe i i contour being chosen so that 7 is positive on it. of integral order n, ' If r is expanded in ascending powers of w, Carlini’s formula is obtained en we approximate by neglecting all powers of w save the lowest,
,{w‘sinhm and when a = O, Cauchy’s formula of §8 2(1) is similarly ob
1 ed by neglecting all powers of w save the lowest, — £111". 1 1r
)=7—rf cos (nﬂ—xsin t9)d6’. 0
y 21183;)(bith being large), it follows from Kelvin,a 5e  t at the important art f th which 0 is small; p 0 8 Path °fn now, on this part of the th
to 6— $03, It IS inferred that, for the vain: of,“ ’ These considerations suggest that it is desirable to examine whether the *4» 1; two terms, namely
‘ — §w2 sinh a — {5103 cosh a, ynot give an approximation valid throughout the first transitional region. 1 n. . . . Y . . .
,gl [0 COS (ng _ $6 + $3763) d6 The Integral which we shall investigate is therefore
1 e [ e—xr d W,
7—r [0 cos (7:6 — x0 + £43.93) d9) ,~ here = — %W‘ sinh a  $W3 cosh a,
. 247—249; . . , f .
see also Emde, A1017“) der Math. und Phys. (3) f‘ It P706. Crumb. phil‘ Soc. XIX. (1918)' pp. 96~110' 1' This is deducible from §8'3l by making a change of origin in the unplane. tY OF BESSEL FUNCTIONS >t an integer, the functions IV(2) and I_,, (2) form
utions of equation (1). y the equation : hm e7“ <2) — 1m] ' 11»); 1/ — 71
§ 3'5) is
<1“ [61—42) _ 6.14:2
2 81/ By m" ' hods of§ 3'5, that Kn (z) is a solution of (1) when been deﬁned, for unrestricted values of 1/, by Z>=1}7T IV(Z.)IV(£)J Sln 1/71' iy be veriﬁed that
T" (2) = lim K, (2). V‘>7L 6) that . (I) , _ . (1) .
m H. (12) = 13719‘5'“ H_,, (12).
18 function K,(z) lies in the fact that it is 3 tends exponentially to zero as 2+ 00 through
.tal property of the function will be established 0 Basset, Proc. Camb. Plzz'l. Soc. VI. (1889),
ven by equations (4) and (5) ; the inﬁnite in
ion will be discussed in 6'14, 6'15. Basset subse
function in his Hydrodynamics, II. (Cambridge, 1888), .v 1 BI_,(2) 31.,(2)
equnalent to 2"“ a” — by "n. ich satisﬁes the same recurr
Treatise on Bessel Function
nition is equivalent to _.,(z) BIy(z)
at ‘ a, p. 11, and
tegrals by ence formulae as I,(z),
a (London, 1895), p. 67, V=n leﬁnition to functions of unrestricted order is by the root v1r{l_y(2)*ly(2)li Math. Soc. xxx. (1899), p. 167. 371] BESSEL FUNCTIONS 79 ' ' ' that
‘ ” ” ‘ ‘ = i ' ﬁ'ers from the serious disadvantage
l 313 S 1 ml) but this function su ' ' MS
(.01. MﬁgAwngeievla; 2v is an odd integer. Consequently in this'worl:i Maciolizaéur—
limzzii): will be used although it has the disadvantage of not satisfying t e sam,
renoe formulae as IV (z). ' ' ' had
An inspection of formula (8) shews that it would have been advaiittlzgeousf if ati‘:;(:re§t: bles
' ' ‘  but in view of the aims rice 0 ex ,
. 'tted from the deﬁnition of Ky (z) . ' ‘ i “he
:eﬁatdldnald’s function it is now inadvisable to make the cdhangie, Sudﬁgcﬁli'5ﬂsggcfisction
' ' f the correspon mg ac or i
is not so undesirable as the presence 0 ’ M rence.
(20:21) because linear combinations of 1,, (z) and If ,, (a) are not of common occur 3‘71. Formulae connected with 131(2) and Xv (2)“ We shall now give various formulae for 1.,(2) and K, (2) analogous tfo
th se constructed in §§3'2——3'6 for the ordinary Bessel functions. The proo s
o .
of the formulae are left to the reader. c , 21/
(1) Iy—i " Iy+i (z) z Iv (Z): Kvﬂl (Z) ‘ KV+1(Z) = _ Ky (Z), (2) I ——i (Z) + 1.14:1: 21»! (Z), KM (Z) + Kv+i = — 21ft, (Z).
(3) 21; (z) + vIu(z)=sz_i (z), szz) + My (2) =— 2K W
(4) 21x2) — v1y(z) =21.+.<z), 2K1 <2)  “113(2): “21"” W ” 'é—lm v ‘ == —m "lewtn(zt
5) (fig) {2"Iv<2>}xz”'“Ivm(zl’ (2012) {21mm < >1: H m IV 1,, m(Z) i‘vn = _ m ’
<6> (it) i Pl“ ’ (aw i z» I < > z (7) IOI(Z)=I1(Z)1 K0/(z)=K1(Z)J I" (Z)=In (z); K—v (Z)=Kv .
The following integral formulae are valid only when R(y + l) > 0 , y i“ ‘ wade
(9) I, (z) = P (y JO cosh (2 cos (9) S111 (ML? ‘1 _2vi h(zt)dt
=Wre>lil ‘) (1&2)’ 1 1 #152 v—geiztdt
= ml —1‘~ ) _ i "eizcososiniv ”Fo+%ﬂWb o _ ﬂ é"cosh (2 cos 0) sin2"9d9
ﬂI‘(v+%)I‘(§)o _ _ ,_2_(iz_):_.f(1  t“)H cosh (2t) dt~
‘ro+5ﬂW»o ...
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