ETNA
Electronic Transactions on Numerical Analysis.
Volume 25, pp. 454-466, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
THE PROPERTIES, INEQUALITIES AND NUMERICAL APPROXIMATION OF
MODIFIED BESSEL FUNCTIONS
JURI M. RAPPOPORT
Dedicated to Ed Saff on the occasion of his 60th birthday
Abstract. Some new properties of kernels of modified Kontorovitch–Lebedev integral transforms — modified
Bessel functions of the second kind with complex order
are presented. Inequalities giving estimations for
these functions with argument
and parameter are obtained. The polynomial approximations of these functions
as a solutions of linear differential equations with polynomial coefficients and their systems are proposed.
Key words. Chebyshev polynomials, modified Bessel functions, Lanczos Tau method, Kontorovich-Lebedev
integral transforms
AMS subject classifications. 33C10, 33F05, 65D20
!
"$#% & '(!
1. Some properties of the functions
and
. In this section
new properties of the kernels of modified Kontorovitch–Lebedev integral transforms are deduced, and some of their known properties are collected, which are necessary later on.
It is possible to write the kernels of these transforms in the form
& )+* '(!, . -!! and "$#/ & '(!0* '(!21.3 -!) 4
where 5 (! is the modified Bessel function of the second kind (also called MacDonald
function).
The functions ) and "$#/ '(! have integral representations [8]
687
!0* 9 -):!;=<?>@BADCFE$G?H/.I CFEJG K I ML I 4
(1.1)
687
"#/ )+* 9 -!:!;=<?>(@NA GPORQH .I G?OQ K I PL ITS
The vector-function UDV) 4 UW$(!? with the components UXV(!+* & ) 4
U W !0* "$#% ) is the solution of the system of differential equations
L W UJV ,ZY L$UJV 1\[ , ]V 1^K W U V , K U W *a` 4
Y
L$ W L$
W _
W
V
L W UW ,ZY L$UW 1 K UJVb1 [ , ] 1cK W UWd*a` S
Y
L$ W L$ W
W _
The functions 'e! and "#/ ! are even and odd functions, respectively
of the variable K ,
f !0* g -)' (! 4
h Received April 28, 2005. Accepted for publication December 22, 2005. Recommended by D. Lubinsky. The
(1.2)
author’s participation in the Constructive Functions-Tech04 Conference has been partially supported by the Russian
Foundation of Basic Research, Grant 04-01-10790 and National Science Foundation, Grant DMS 0411729.
Russian Academy of Sciences, Vlasov Street, Building 27, Apt. 8, Moscow 117335, Russia
(jmrap@landau.ac.ru).
454
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MODIFIED BESSEL FUNCTIONS
455
"$#% )+*i1 "$#/ -)' (! S
The functions f & ' (! and "$#% & (! are related to the modified Bessel functions of the first kind "&5 ) as follows,
f" " & '(!
(1.3)
& 'j(!+* CFEJGPH k Kj - -)'(!l1.
4
k
"$#%" "$#/" ' (!
(1.4)
"#/ )+* CmE$G?H k Ke - -! !21.
S
k
The expansion of " ) in ascending powers of has the form
" & ' (!+*on .ep n CmE$G n&Krq Q .lp , 3 G?OQ nmKsq Q .eptp
w
:] p
7
v
v 7 w 3M„ w
n
u
4
(1.5)
wTx 9 ytz {}|~y ,€W , 3 Kj * wTx 9 ƒ‚ ,
„
where ‚ w and w satisfy the following recurrence relations:
CmE$G | Krq Q :W  , 3 GPORQ | Krq Q :W 
9‚ , 3M„ 9 *…n . p 4
{ | , 3K 
W
y V
# w *† W ‡ y | y , V W W W 4 ‰ w *Š W ‡ y | y K V W W 4
n , W  ,ˆK p
n , W  ,ˆK p
w‚ *a‚ w - V # w , „ w - V ‰ w 4 „ w * „ w - V # w 1‹‚ w - V ‰ w S
The expansion of " - g -)'j(! in ascending powers of has the form
- " - -!!0* n . p n CFEJG n Ksq Q . p 1 3 GPORQ n Krq Q . ptp
w
:n ] p
7v
v7 w 3 w
u
4
(1.6)
)
y
z
{
y
|
3
wTx 9
, WV 1 K  * wFx 9 (Œ , L
where Πw and L w satisfy the following recurrence relations:
CmE$G | Krq Q:W 1 3 GPORQ | Krq Q:W 
9Œ , 3 L 9 *…n . p - VPŽW
4
{}| V 1 3 Kj
W
y
‘ w *† W ‡ y | y 1 WV
4 “ w *† W ‡ y | y K V W W 4
W
V
W
n 1 W  ,’K p
n 1 W  ,’K p
wŒ *aŒ w - V ‘ w 1‹L w - V “ w 4 L w *†L w - V ‘ w ,”Œ w - V “ w S
The expansions (1.5) and (1.6) converge for all `–•8}•†— and `™˜šK‹•†— .
It follows from (1.1)–(1.2) that it is possible to write f ! in the form of the
Fourier cosinus-transform
2 ›eœž -!:!;=<?>(@NA CFEJGPH I
)+*…n k . p
(1.7)
.lŸ I¡ K¢ 4
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456
and
J. M. RAPPOPORT
"$#% )
(1.8)
in the form of the Fourier sinus-transform
"#/ & )+*on k . p
0 ›e£’ )- :N;=<Ž>@!AJG?ORQH}I
.¡Ÿ Il K¢ S
The inversion formulas have the respective forms
e› œ†¤
f ! Ÿ K ¥
* n k . p -):N;=<Ž>@!AJCmE$G?H¨.I 4
I~¦ §
›e£c¤
)- :N;=<Ž>@!A G?ORQH}I
"#/ ! Ÿ K ¥
p
.
.
*
§
n
k
I~¦
or, in integral form,
6 7
9 6 7
# 9 "/
-!:B;=<Ž>@BA FC EJGPH .I 4
(1.10)
-!:B;=<Ž>@BA ?G OQHª. I S
Differentiating equations (1.9) and (1.10) with respect to I , we obtain
687
G?OQ
G?OQH I CmE$G?H/. I 1 G?OQHª. I -!:B;=<?>@!A 4
_
9 K & ) I KjPLK^* k . [D
6 7
CmE$G I KjPLK^* k . [ CFEJGPH .I 1^ GPORQH I G?OQNH . I -!:B;=<?>@!A S
(1.11) 9 K "$#/ & '(!
_
(1.9)
It follows from (1.9) that
and from (1.11) that
'e! CFE$G I KeML$Kc*©k .
'(! G?ORQ I KeML$Kc* k .
6 7
!- :
9 f !PLKc* k . 4
6 7
-):
9 K "$#% & )ML$K^* k . S
.
Differentiating (1.9) and (1.10) ‰ times with respect to I , we obtain
6 7
-!:!;=<Ž>@BA CFEJGPH¨.I 4
CmE$G
_
9 K W?« ' (! I KjPLKc* k . M1 Y «D¬AWŽ« [ 6 7
-!:!;=<Ž>@BA G?OQH .I 4
G?OQ
_
9 K W?« "$#/ & '(! I KjPLKc*­k . M1 Y « ¬ AWŽ« [ from which there follows, for I *†` ,
6 7
-):!;=<?>@!A CFE$G?H .I
_ Ax 9 S
9 K WŽ« & j(!ML$Kc*­k . M1 Y « ¬ AWŽ« [ .
Differentiating (1.9) and (1.10) ‰ ,
times with respect to I , we obtain
Y
687 -):B;=<Ž>@!A CmE$G?H .I 4
G?OQ
_
9 K WŽ« V f V?ŽW ' ) I KjPLKc* k . M1 Y « VF¬ AWŽ« V [ 6 7 -):!;=<?>@!A GPORQH .I S
CmE$G
_
9 K WŽ« V "$#% V?ŽW ' (! I KjPLKc* k . M1 Y « ¬ AWŽ« V [ ETNA
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MODIFIED BESSEL FUNCTIONS
457
I *a` ,
6 7 WŽ« V -!:B;=<?>@!A GPORQH .I _ S
9 K ?W « V "$#% VPŽW ' (!PLKc* k . M 1 Y D« ¬ A [ Ax 9
For the computation of certain integrals of the functions f & '(! and "$#% & '(! ,
whence, for
integral identities are useful. They reduce this problem to the computation of some other
integrals of elementary functions.
P ROPOSITION 1.1. If is absolutely integrable on
, then the following identities
hold,
®
¯ ` 4 —”
6 7
6 7 -):N;=<Ž>@!A CmE$G?H¨I ›jœ
®
p
.
. I ML I 4
&
'
(
!
(
j
K
ML
$
^
K
o
*
n
k
9
9 6 7
6 7 -):N;=<Ž>@!A G?OQNH}I › £
(1.13)
$
"
#
/
®
p
.
. I ML I 4
n
&
'
(
!
(
j
K
ML
$
^
K
*
k
9
9 ›eœ
›¡£
where
I is the Fourier cosinus-transform of ® Kj , and I the Fourier sinus-transform
of ® Kj .
Proof. Multiplying both sides of the equalities (1.7) and (1.8) by ® Kj , integrating with
respect to K from ` to — , and applying Fubini’s theorem for singular integrals with parameter
[22], we obtain (1.12) and (1.13).
P ROPOSITION 1.2. If ® is absolutely integrable on ¯ ` 4 —” , then the following identities
hold
6”7
›œ
6”7 -!:!;=<Ž>@!A CmE$G?H¨I
(1.14)
p
.
. ® I PL I 4
n
(
!
j
K
PL
c
K
*
&
'
k
9 9
6”7
›l£
6”7 -!:!;=<Ž>@!AJG?OQNH}I
$
"
%
#
(1.15)
p
.
. ® I PL ITS
)
j
K
PL
c
K
o
*
n
k
9
9 (1.12)
Proof. This follows from (1.9)–(1.10) and from Fubini’s theorem [22].
The equations (1.12)–(1.15) are useful for the simplification and the calculation of different integrals containing
and
.
VPŽW (! ›eœ "$#% VPŽW ' (! ›e£
For example, let ® Ke+* -)°g , then
I +*€± ²W ° °tA , I +*€± ²W ° tA A and
6 7
-t° LKª*´³ 6 7 (³ W , I W - V -):N;=<Ž>@!A CmE$G?H¨. I L I 4
'
(
!
9
9
6”7
687 -)°A~-!:B;=<?>@!A CFE$G?H¨I
. LI4
9 & ) ³ W ,ˆY K W LKª* . k ³ 9 687
”
6
7
-t°
- -):!;=<?>@!AXGPORQH¨.I L I 4
9 "#/ ' (! LKª* 9 I (³ W , I W V 6 7
6 7 -)°A~-!:!;=<?>(@BA GPORQH¨I
K
$
"
%
#
.
. L ITS
&) ³ W ,ˆK W LKª* k 9 9
{ ]V , { ]V 1 , then ›eœ I 2* µ W ² and
If ® Ke2*
W
W
;=<Ž>@BA
6š7
{ ‡ Y , 3 . K { ‡ Y 1 3 . K PLKc*
)
9
6”7
CFEJGPH A
*ž¶ . ktk 9 -!:!;=<Ž>@!A ¶ CmE$G?H W I L I * k ¶ k -¸· 9 . S
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458
If
J. M. RAPPOPORT
›l£
® (Kj+* ;=<Ž>@» W >¹² ºm$@¼» tW ² ;=<Ž$>~¼ » W ² °¼ , ½ f ³ ½ • WV , then
687 "#/ '(! G?ORQH . Kj
6”7
9 CmE$G?H . Kj, CFE$G . k ³e LKc* .Y 9
k
k
I +* µ =; W <?² >@>(»R¹ º&°@A¼ ¾ and
-!:!;=<Ž>@!A CmE$G?H (³ I ML I * .Y ° ) S
R EMARK 1.3. All formulas of the present paragraph remain valid if
lying in the right-hand half-plane.
is changed to
¿
' (! "$#/ ' (!
f !
CFEJG K I CFEJGPH .I 6 7 -»ÂÁmt;=<Ž>@NA¼(: LÃL I
9
CFEJG (K I CFEJGPH/WA ‘ CFEJGPH
³l
CFEJGPH I , CmE$G?H ³ L I *
CmE$G?H A
CFEJG
›œ
[ CFEJGPH I , CmW E$G?H ³ _ * k . CFE$G?HÅ° Cm(E$³jG?KjH jK S
W
k
1.1. The Laplace transform of
and
.
The Laplace transform of
is computed in [21]. We use the representation (1.1) for the
evaluation of the Laplace transformation of
. We have
' (!
À ¤ '(! Ÿ K ¦ * 6 7
9
6 7
* 9
*ۀ k .
Equivalently, we can write
ÉÈ
- V mC E$G?H
À - VÆÇ CFEJG (K FC E$G?H - VJ‘ o
+
.
p
*
n
k Kj f ) S
k
± Ám W V
For the evaluation of the Laplace transform of "$#% & ) we utilize the representation
(1.2). We have
G?ORQ (³jKj
À ¤ "#/ ' (! Ÿ ‘ ¦ * Ä k . l› £ [ PG ORQH WA
.
CmE$G?H I , CFJE GPH ³ _ * k mC E$G?H Kj G?ORQH°W 4
k
or, equivalently,
ÉÈ
À - VÅÆÇ G?OQ (K FC EJGPH - VJ‘ * Ä k . CmE$G?H Kj "#/ ) S
k
± Á- W V
We note that these equations can also be obtained directly from the formula for the Laplace
transforms of
by separating real and imaginary parts.
5 !
1.2. The asymptotic behavior of
. For
and
are valid [8],
K —
K —
and
'(!
f !bʧn
"$#% !bʧn
where
is a fixed positive number.
f & )
"#/ )
"#/ !
` 4 —
and
for
the following asymptotic formulas for
pk -0ËFªÌ CmE$G &n Krq Q K¨1cK/1‹Krq Q
K 1cK¨1^Krq Q
k p - ËF Ì G?OQ nmKsq Q /
. p 4
. p 4
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459
MODIFIED BESSEL FUNCTIONS
` we have
- mC E$G | Krq Q :W  1 3 G?ORQ | Ksq Q :W 
!0ÊÍ.Y n . p
4
{ | V 1 3K 
W
It follows immediately from (1.3)–(1.6) that for
whence
- % Î {
'(!+Ê .Y n . p
[Ï.Y , 3 K _
, "$# { [¸.Y , 3 K _ G?OQ mn Krq Q
- Î {
"$# [.Y , 3 K
"#/ ' (!+ÊÑ.Y n . p
CmE$G n Krq Q . p
_
.lp!Ð 4
CmE$G n&Krq Q . p
1 { [ .Y , 3 K _ ?G OQ n Krq Q . p)ÐÅ4
For large values
where
the following asymptotic expansion is valid [9]
)- : v 7
w
& ' (!+Êon . k p wTx 9 Ï[ .Y , 3 K 4 y _ . ! - 4
‡ Ò W 1 W m ‡ Ò W ‹
W )Ô&ÔmÔ ‡ Ò W 1š . y 1 W
y
1
Ó
4
.
ƒ Ò 2*
Y
Y S
W w y)z
In particular, therefore,
)0ʧn . k p -!: [ Y 1
"$#% )0ʧn . k p !- : [ . K K. W †
, Ô&ÔmÔ _ 4
,aÔmÔ&Ô _ * . K n . k p -): Y ,†Ô&ÔmԎ S
1.3. The series expansions in powers of K . The solutions of problems in mathematical physics connected with the use of the Kontorovitch–Lebedev integral transforms are
often expressed as integrals with respect to K of the functions ) , & '(! and
"#/ 'e! . Both the asymptotic expansions of these integrals for large values K , and the
expansions of these functions in powers of K , are of interest for the analysis of the behavior
of these integrals.
The expansions of these functions in powers of K are deduced from their integral repreCmE$G (K I and G?OQ (K I by their series expansions
sentations (1.1)–(1.2). Substituting in them
and interchanging the order of the summation and integration, we obtain
v 7 K W w 6 7 W w -):!;=<?>@!A
' (!+* wTx 9 . y z 9 I LI *
W 6 7 W -):N;=<Ž>@!A K ] 6 7 ] -!:B;=<?>@!A
K
9
(1.16)
* (!l1 . z 9 I LI, ‡z 9 I L I ,aÔmÔ&Ô 4
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460
J. M. RAPPOPORT
v 7 K W w 6 7 W w -!:!;=<?>(@BA CFEJGPH¨I
' (!+* wTx 9 . y z 9 I .LI *
W 6 7 W -):!;=<?>@BA CFE$G?H/I
K
.LI
* !¡1 . z 9 I ] 6 7 ]
(1.17)
, K ‡ z 9 I -!:!;=<Ž>@!A CmE$G?H . I L I ,aÔmÔmÔ 4
v 7 K W w V 6 7 W w V -!:!;=<Ž>@!A G?OQNH}I
" #/ '(!+* wTx 9 . y , z 9 I .LI *
6 7 -):BY ;=<Ž>@!A G?ORQH¨I
6 7
(1.18)
. L I 1 K Ó  z 9 I  -):N;=<Ž>@!A G?OQNH}.I L I ,aÔmÔ&Ô S
*aK 9 I K
These functions are entire functions in the variable , and therefore the series converge
for all real values of . From these expansions it is possible to obtain the series for the
derivatives and for the integrals of these functions with respect to the variable , which will
converge for all real also. Similar integrals for the spherical functions are stated in [23].
It’s possible to rewrite the expansions (1.16)–(1.18) in terms of Laplace transforms as
follows,
K
K
K
(1.19)
(1.20)
(1.21)
v7
)
:
' (!+* wTx 9 M 1 Y
v7
)
:
& j(!+* wTx 9 M1 Y
v7
)
:
$" #/ & (!+* wTx 9 M1 Y
w À€Õ&Ög×Ø CmCmE$G?H W w U,
Ÿ
U, wY W 1 Y Y U w À Õ Ög× CmCmE$G?H W U,
Ÿ
.
¶ wU Y U w À Õ Ög× CmCmE$Ø G?H W V (U,
. (U, . Y Ÿ U
Ww
K
y
D Ù . z 4
Ww
K
Ù . y z4
Ww V
K
Ù . y , z4
Y
This form of writing may be more convenient since it is possible to use numerical methods
for evaluating Laplace transforms.
The expansions (1.19)–(1.21) are convenient for the calculation of the kernels of Kontorovitch-Lebedev integral transforms for small values .
K
) , f j) and "$#% ) .
¯
4
K‹Ú ` 8
—
½ f ) ½ ˜ (!+*on . k p -!: 4
and it follows from (1.2) that for all K‹Ú ¯ ` 4 —8
687
-):
¤
½ "$#% ) ½ ˜ 9 -):N;=<Ž>@!A G?ORQH .I L I * n . k p : Y 1’ÛeP . ! ¦ ˜ŠÜ 4
where Ü is some positive constant [10],[11].
3
In [4], for arbitrary ҏ*†Ý, K , ݪފ` , the following inequality is derived
½ "m5 (! ½ ˜ Ëß Ìß "Fà ) S
2. Inequalities for the MacDonald functions
It follows from (1.1) that for all
Taking advantage of the formula [4]
½ Å5 ) ½ ˜ ná V ( 4 Ýe, á W 4 Ý ½ K ½ à - p - Ë ß Ìß 4
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MODIFIED BESSEL FUNCTIONS
461
âd4½ K ½Xãšâ ,
½ ' (! ½ ˜ á ! - Ë ß Ìß S
we obtain that beginning with some
But this inequality is too rough and may be insufficient for conducting various proofs. To
obtain more refined inequalities, we use [5]
(2.1)
where
ä
½ ) ½ ˜Šä - å - Ë ß Ìß 4
is some positive constant, and the representations [8]
-):
)+* n . k p CmE$G? H Kj
QH k 6 :  -»R:g-)çF¼ -»:çF¼
(2.2)
, KÃæ Ö . k Kj 9 ¶ 1^U 1 ¢ ' (UDPLU
K1 Ãæ Ö . Qk H k Kj 6”7 -»R:& çm¼ U L$U 4
U
k687 -)ç : :
'
(U
"$#% )+* .K :
k
UtU–1c) LU S
L EMMA 2.1. The following inequalities hold for ã `
. -!: - ²Né é
g
Ë
ß
Ì
ß
Ï
å
è
4
½ ' ! ½ ˜šŒ ½ K ½ , [ k _ ½ "$#% & ) ½ ˜šŒ 9 ½ K ½ - Ëgß Ìß - åè 4
where Π9 and Πare some positive constants.
Proof. We estimate the second additive term in (2.2), using the inequality (2.1),
(2.3)
6 :  -»:-!çF¼
QH
½ Kræ Ö . k jK 9 k
Å1cU -!: 6 : ˜šä ½ K ½ - Ëß Ìß ¶ 9 ¶ Vê=ë
We next estimate the third additive term,
(2.4)
-»R:&çm¼
1 ¢ U U L$U ½
ç
· 1 U - å LU˜8ä ½ K ½ - Ëgß Ìß ¸- åè S
U
ë
6 7 - »:çF¼
8
QH
-!: 687
½ Kæ Ö . k Kj : U U $L U ½ ˜ŠÜ ½ K ½ - Ë ß Ìß : !- ç U 0- åì L U
¶ k
¶
¶
Ë
ß
Ì
ß
- W : -båè S
˜ŠÜ ½ K ½ Combining the first term and estimates (2.3) and (2.4), we obtain the required inequality.
Furthermore, we obtain
: 6 7 -)ç U
K
½ "#/ 'e! ½ ˜ ½ . 9 U U™1c L$U ½
¶ k
¶ 6 «í Aç -)ç - åì
U L$U؊Œ 9 ½ K ½ - Ë ß Ìß - åè S
˜šŒ 9 ½ K ½ - Ëß Ìß : :
¶ Uî1^
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462
J. M. RAPPOPORT
%à ' (!
for large
For future use, an analysis of the behavior of the modified Bessel function
values of is necessary.
L EMMA 2.2. For
,
,
, the following inequality holds,
K
`–˜šÝc˜ WV ½ K ½ Þ8K 9 Þ Y ªÞ” 9 Þ Y
½ à ) ½ ˜ n Œ&VT à - ,”ŒFWF g - à ½ K ½ à - p :g- gË ß Ìß 4
where Œ&V ã ` , ŒmW ã ` , Œ&V , ŒFW , K 9 , 9 are some constants.
Proof. We use the formula [6]
ï (!+* . G?OQ k " - ï (!¡1 "Fï (!P 4 ð *´Ý, 3 K S
ktð
» V ¼ ã ` , K 9 » V ¼ some
1. We first estimate G?OQ . It is possible to show that for ½ K ½ ޞK 9
t
k
ð
constant, the following inequality is valid
(2.5)
‚ V ²Né é ˜ ½ G?OQ ktð ½ ˜Š‚ W ²é é 4
where ‚ V ã ` , ‚ W ã ` , ‚ V , ‚ W are some constants.
3
2. We next estimate " ï ) , *ñÝÅ, K , ݈Þò` . The following inequality is derived for
ð
Ýcޚ` in [4]
: à Ëß Ìß
½ "mï ! ½ ˜ " 9 ! { W ƒÝó, S
Y
» V ¼ ã ` , 9» V ¼ some
It follows from the asymptotics of " 9 (! [10] for large that for ŠÞ€ 9
constant,
½ " ï ! ½ ˜Š‚  à - : ː ß Ìß 4
(2.6)
where ‚ ã ` 4 ‚ some constant.
3


3. We finally estimate " - ï (! , *òݏ, K , ݋ކ` . Proceeding analogously [4], we can
ð
rewrite " - ï (! in the form
: -ï
" - ï !0* { W 1 ô ( 4 ð 4
Y ð
õ
where
õ: W
võ 7
wFx W
ô ( 4 ð +* Y , x V ö zT÷ wFx V P1 , y S
õ Vð y .
õ
Ø y
y
y
W
W
½
4
4
Then ½ 1
for `ø˜8
,
,
and
*
‹
1
Ý
ˆ
,
K
Þ
1
^
Ý
˜
*
Ó
m
S
m
S
S
W
Ø
Y ÷ wTwTxx Ø y 1^Ý W ,’K W Þ » - V ¼ƒù . We obtain,
Y 1‹Ý W ð ,ˆK W Þ WV 4 K arbitrary. Therefore,
W
õ V
after some calculations, that
võ 7 : W W
.
ô 4 𠸘 Y , x ö z ö 1 z ˜ . n Y , . " V! p S
V
Y
»
¼
W
V
9
Using for ½ K ½ ÞñK
from [5]
Þ Y , `ª˜€ÝŠ˜ W , the expansion of the gamma-function
»
¼
»
¼
W
W
9
9
and the asymptotics [6] for " V (! we obtain that beginning with some , }ޚ
Þ Y , the
following estimation holds,
(2.7)
½ " - ï ! ½ ˜Š‚ ] $ - à ½ K ½ à - :& Ëgß Ìß 4
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Kent State University
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463
MODIFIED BESSEL FUNCTIONS
‚ ] ã ` , ‚ ] some constant.
Combining the estimations (2.5)–(2.7), we obtain that for `c˜\Ýò˜ WV , ½ K ½ ÞúK 9 Þ
,
Y
»
¼
»
¼
»
¼
»
¼
V
W
V
W
9
9
9
9
9
9
9
¨Þ8 Þ Y , K *Šû Öü K 4 K , *†û Öü 4 the following inequality is valid,
½ ï )¸˜ . k ‚ V - ²é é ½ " - ï (!l1 "Fï (! ½ ˜ . k ‚ V n ‚  à - ,”‚ ] $ - à ½ K ½ à - pf :- Ëß Ìß S
²
²
Denoting Œ V * Wþý ‚ , Œ W * Wþý ‚ ] , we obtain the statement of the lemma.

The properties of the modified Kontorovitch–Lebedev integral transforms are considered
in [7]–[15].
3. Tau method approximation for modified Bessel function of imaginary order.
Several approaches for the evaluation of the modified Bessel functions are elaborated in [1]–
[2]. The Tau method [3] realization, with minimal residue choice for the determination of the
polynomial approximations of the solutions of the second order differential equations with
polynomial coefficients [16] of the following form
(‚ 9 U W ,”‚XÿFU U,´(‚ V Ur,ˆ‚ W (UDj,ˆ‚  )U2*´` 4 !ƒ`$+*a‚ ] 4 UÅÚ ¯ ` 4 Y 4
is supposed. An ‰ -th approximation of the solution is sought in the form of the ‰ -th degree
polynomial « U , which is the solution of the equation
„ 9 U W , „ ÿ U!(UD2* 6 ç „ VTs, „ WFU, „ !(!PL–, „ ] U,m« W â ¯ 1c³j« WMUf,‹³e« W 4
« W Y
9

3
„ 3
where the coefficients ‚ , *…` 4 S&SmS 4 , may be expressed by coefficients , *o` 4 SmS&S 4 ,
²
?
G
R
O
Q
W
] » « W ¼ — the leftmost root of the shifted Chebyshev polynomial of the ‰ , . -th
³j« Wd*
degree â « W (U in the interval ¯ ` 4 , &« W — undefined coefficient.
Y of the polynomial « (UDÏ* «wTx 9 ‘ w U w , which is the
The problem about determination
least deviated from zero on the interval ¯ ` 4 among all ‰ -th degree polynomials, satisfying
Y
« »« ¼
the pair of linear correlations on the coefficients ‘ 9 * ` , x V Œ ‘ *
Y was considered.
The following theorem is proved [16]:
» « ¼ 3 4 SmS&S 4P‰ , is alternating, then the
T HEOREM 3.1. If the sequence
of numbers Π, *
Y
polynomial &« â « ¯ 1}³e«N=Ud,‹³e« is the polynomial least deviating
from zero in the uniform
Y
metric on ¯ ` 4 among all polynomials of degree ‰ , satisfying the indicated pair of linear
Y
relations.
On the basis of this theorem it’s shown (as suggested by us) in the Tau
method residue, in
, among all possible
a number of significant cases, is a minimal in the uniform metric on
polynomial residues permitting the Volterra integral equations solution.
We have the following differential equation with polynomial coefficients for the approximation and computing of the second kind modified Bessel function
:
¯` 4 Y
' (!
U W (U, . Ur, Y (UDj,a Y ‡ ˆ
, K W )U+*†` 4
!ƒ`$2* Y 4
and the Volterra integral equation
6 ç  U W !(UD0* 9 [ ‡ ,ˆK W _ 1\[ ‡ Y ,’K W _ Uî1 . ¢)!PLó, . U S
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Kent State University
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464
J. M. RAPPOPORT
wTx 9 “ w U w
.
.
“ 9Ž9 * ]V W 4 “ 9 w i
y
* 1 W,
,ˆK
We
obtain the following
recurrence formulas for the coefficients of canonical polynomials
in this case:
U2*
y .
“ 9 w 4 y * Y 4 S&SmS
y , , ] V ’
, KW -V
é é *
The minimality
of the residue suggested by us follows from the Theorem 3.1 as ,
The advantages of this modification, as compared with usual and other tau-methods, is
shown.
M1 Y
# *´` 4 Y 4 &S SmS
4. Tau method approximation for modified Bessel function of complex order. A
new numerical scheme of the Tau method application is proposed for the solution of the second order linear differential equations systems, with the second order polynomial coefficients
of the following kind:
w
v
?¼ W
»
» ?¼
(‚ 9 U ,”‚ V U (U, x ¯ (‚ »?¼- V U™1^‚ »? ¼ (U,ˆ‚ »?¼ V U *a` 4
V
y 4 UÚ ¯ ` 4 4
»?¼
4
4
(`J2*a‚ w W 4 *
&
S
m
S
S
Y
Y

in the unknown vector-function )Ur*ZXV(UD 4 S&SmS 4 w U? . It is assumed to have only one
solution. Integrating twice and carrying an addition in the right part in the kind of the vectorpolynomial «U , we derive for the determination of the ‰ -th approximation of the solution
)U+*ñ V (U 4 S&SmS 4 w (UD? the system of Volterra integral equations with polynomial kernels
w
„ 9» ?¼ U W , „ » ?¼ U (UD2* 6 ç ¯ v „ » ?(¼- ™, „ » ? ¼ Ur, „ » ?R¼ (! L$ø, « W (U 4
V
9 x V  V

 V
*
4Y SmSmS 4 y 4
y . and * 4 S&SmS 4 y , are connected in
„ » ?¼
» ?¼ 3
where the coefficients and ‚ 4 *§` y 4 SmSmS 4 Ó ,
Y
4Y SmSmS 4 4 are ‰ , . -th degree polynomials.
a definite way and « W U , *
The different
variables of the vector residue choice and its minimization are analyzed. The recurrence
formulas for the canonical vector-polynomials coefficients convenient for the calculations are
given.
Consider the system of two second order differential equations
in more detail.
This case is of particular interest for differential equations with complex coefficients.
The scheme of the integral form of the Tau Method described in this paper can be used for
deriving polynomial approximations of hypergeometric and confluent hypergeometric functions of the first kind with complex parameters.
The modified Kontorovich–Lebedev
integral transforms [7] with kernels
y * .
õ
»
:
¼
R
»
&
:
¼
R
»
&
:
~
¼
»
&
:
¼
& )+* !#" Ì W %$%" Ì and "$#% & '(!+* &%" Ì W '$%" Ì , where )
is MacDonald’s function, is of great importance in solving some problems of mathematical
physics, in particular mixed boundary value problems for the H ELMHOLTZ equation in wedge
and cone domains. We find it necessary to compute
and
to use this transform in practice [13]. These functions also
occur in solving some classes of dual integral equations with kernels which contain MacDonald’s function of imaginary index
[7]. Therefore, now we consider the second kind
modified Bessel function
in more detail.
& )
"#/ !
' (!
)
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MODIFIED BESSEL FUNCTIONS
465
We have a system of two second order differential equations
UW V ,
UW W ,
. (U, ˆ
W
Y V , K $V2,ˆK(gWÃ*´` 4
. (U, ^
W
Y W 1 K($V0,’K gWÃ*´` 4
$Vg(`$2* Y 4 gWƒ`$2*´` 4
or the system of Volterra integral equations
6
6 ç
.
WU V (U2* 9 ç P . ’
W
W
, K =Å1š ,’K UP V )ML$ø,ˆK 9 1^U W (!PLó, . U 4
6
6
WU gWJU2*aK 9 ç (Uî1^!$V(!PLó, 9 ç P . ,’K W =Å1š . ,’K W UPgW$(!ML$ 4
)+*€ . k )- : V X Y , 3 W X Y 4 ¨Þ Y S
The following formulas for the coefficients of canonical vector-polynomials are derived [16]
»“ V ¼ * (K W , # # , Y Pm # , Y F # , . 4 “ » V ¼ * K+ # , . F # , Y 4
W
V
K W , # # , Y ? W ,ˆK W
(K W , # # , Y ? W ,ˆK W
“ V » W ¼ *€1 “ W » V ¼ 4 “ W » W ¼ * “ V » V ¼ 4
»“ ? ¼ *i1 . 3 , Y FP(K W , 3 3 3 3 , Y ? “ V»?R¼ V 1^K “ W »?R¼ V 4
V
K W , , Y P W ,’K W
. 3
» ?¼
»?¼
W 3 3
“ W »? ¼ *i1 , Y FK K “ WV , V 3 ,a
3 ,(K P, W ,’ K, W Y ? “ W R V 4
Y
3 * # 1 4 S&SmS 4 ` 4 * 4 . S
Y
Y
By means of computations is shown
that
the
choice
of
the
residue in the form « W Ur*
.
â
¯
4
,
,
is
optimal
as
compared
with other known variants
« W « W 1–³ « W MUl,ų « W
* Y
Y
in this case too.
The applications for the numerical solution of boundary value problems in wedge domains are given in [18],[19].
Acknowledgments. The author wishes to thank the organizers of the Constructive FunctionsTech04 Conference for their encouragement and support.
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ETNA
Kent State University
etna@mcs.kent.edu
466
J. M. RAPPOPORT
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0/
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m