Evaluation of incomplete cylindrical functions

EVALUATION

OF INCOMPLETE CYLINDRICAL

FUNCTIONS*

M. M. AGREST Sukhumi (Received

5 November 19681

1. Connection between the incomplete cylindrical functions and the incomplete Weber and Lipshch~tz-HankeI integrals The incomplete cylindrical Poisson form

E: fw* 4=

s

l”exp(f

(ZP)” r

functions (ICF) were introduced in [l] in the

ig co.9t) sirPt

(Y + ‘1%)I? (l/e) ‘o

dt,

WY

-k W>O, w

the Bessel form ul

..(w,z,=fSexp (psh t -

vt) dt

(1.2)

0

and the Sonin-Schlbfli form

where v, 10, p, q and z are arbitrary complex variables. The ICF’s are particular solutions of the differential VVY

Lzz

Py zadz"

t

.z.%az

+

(2” -

Y2) y = f

equations

(2; y, 4,

where the T(r; I’; u-‘)are analytic functions of the variables z, v and u: and

41

(1.4)

of diff.

equatim

v-zchw)erp(zshw-vvw)-(v+z)]

R&S.

zzB,(w,

z)=

sz

t

J

I

I [

_--

z+v z

0,

w=S,

2-v x

B, iwt 2)

(z-v)cosvlt ?I

sin vit,

= n)

cj

’

w = in

ia

a f&p_

w=

w=ioo

of w, p.

form of r.h.s. vab

w = 2iqv

0,

pPrticular

Particular

SCHEME

4

functiofm

Weber

functions

Auger

Cylindricel function6

stnrcturs functiona snd modificsztionr

E,*@,

Solution of diff equetion

. a

f : B

T--

class of ICF -

I

Y”fi

Lomme I u, y,

wt

P,

2)

2)

v=f&

[+(a,

Function8

Weber

nflJe,(w),

Lipschitz-Hankel incomplete integmle 2)

I

a

?

Evaluation

identically

vanish

generalization Basset, ICF’s

of incomplete. cylindrical

for certain

values

of the cylindrical

Hankel

and MacDonald

are also encountered

functions

of w (see Scheme).

functions; functions

as Struve,

43

The ICF’s

the classical

Bessel,

are particular

cases

Anger and Weber etc.

are thus a Neumann,

of ICF’s. functions

The (see

Scheme). The domain

of variation

of the variables 0

Re (v + %) > 0, is fundamental be continued ([ll,

<

Re u; <

for the ICF in the Poisson analytically

outside

form.

the region

--n
a-r,

(1.5)

The functions

(1.5) by means

of this class

of reiluction

can

formulae

pp. 36-37).

Notice that the reduction formula given in [ll with respect to the variable when w is not an integer, only holds for the lower half-plane, where Im w < 0. This was pointed out to the author by N. Ya. Viletskin, In the general case, the reduction formula is

E: (w, Z) = 2

sin kvn

-Jy

,-i (k- 1) yx,

(4x

ee+i(k-l)vx,

1m

w

to whom many thanks.

u)

>

0

+ lmw<~

+ E$(w-kkn,z)

x

e-2ikvx,

Im 10> 0,

e+2ikvn, Im w < 0, 0<

where kn, z)

E: (w - kn, z) on the right-hand

Re(w-kkx)
to even k, and EJ (IL. -

side corresponds

to odd k.

Inside the fundamental region (1.5), the set of ICF’s the set of incomplete Lipschitz-Hankel integrals

(1.1) is equivalent

to

(1.6)

Ze, (a, 2) = \ e-at i?Z, (t) dt,

I, where 2” (t) is a cylindrical Any pair means (1.6).

Bessel,

Neumann,

Basset

or MacDonald

function.

E,+I(w, z), E:+, (w, z) of the set of ICF’s (1.1) may be mapped by

of a linear nondegenerate transformation In particular (111, pp. 52, 54),

into a pair of functions

of the set

M. M. Agrest

44

-i co9 w.

a=

We can similarly establish a connection between the set of ICF’s f1.2) or (1.3) and the set of incomplete Weber integrals Q,, (5, z) , Qy (3, z), Pv (2, z) and P, (2, z), where Qy (r, z) == (23.)‘1s

tV+lJV(t) exp (Z --- ;)

dt,

(f.9)

0 mexpih

P, (5, z) := (22$-“-~

Rev >

-1,

S

tv+lN.+ (t) exp

z 12h--

ar g z ,!

x -

-5.

dt,

@Xl)

n ,2&) 9 3*

and the functions 0, fx, z) and Py (x, z) are defined by the same integrals after replacing Jv 01 by I, (t) and N, (t) by K, (t) respectively. When the ICF index is fixed but arbitrary, in the Poisson or Bessel form, and is regarded as a function of the two complex variables w and z, one form cannot be expressed al~braically in terms of the other, since they belong to different classes of functions (tll, p. 151). The same applies to the Lipshcitz-Hankel incomplete integrals (1.6) and the incomplete Weber integrals (1.9) and (1.10). However, if the index v = n is an integer, linear relationships exist between the functions of the different classes, and between members of a given class, The following relationships will be used below (111, pp. 91, 146): sn(~,

z) = qPn(w, z) +*/‘z[En+(n12,

z) -J&+(71/2+

iw, z)]

n =O,

1;

Evaluation of incomplete

cylindrical

45

functions

TABLE 1 Range

iTabulated

NO.

14

a Incomplete functions:

Bessel

Jn(a, P), HR(a, Modified ICF’s: Fn’@,

16

p)

0(0.01)1.57;

P) (n=O,i);

Is&a, Kso(a,

P).

W(a,

P)

P).

eapIeo

(a,

p), P),

Stnwe functions of them: WP),

;

0.1(0. I)10

“$2

1.10-e

(n=O,1)

Incomplete modified Hanks1 intearala: XaEKenfr we+

59

and Stnwe

functions: Jet+,

of D

1.2.10-8 0.05 0.05)50 when a = 0 0.05 I0.05)10 when 1%I< 2.1 1.2.10-7 0.05(0.05),,,%henIaj>2.!

Lipscbitz-Hankel

Incomplete

+Jetl(%

60

of variation

function8

P)

0(0.1)10 0(-O. i)(-10)

Lipachitz-

O(0. I)10

ewp Jc0@, Kco(a,

P)

0.05(0.05)10

when a (2.5

1.2.10-8

0.05(0.05):

when a > 2.5

1.2.10-7

P)

and integrals oy).

E&J)

0(0.001)5(0.005)15(0.01)100

I* IO-‘1

9

The function c&,(w, z) in (1.11) is easily expressed tabulated functions; cpO (ui, z) = 0.

in terms of well-known

When u = n is an integer, there is also an algebraic connection between the ICF class memebers Je,,(a, z), Nev(a, z) and the Lommel functions U, (w, z) and Yv (w, z), called in the literature, e.g. [21, cylindrical functions of two variables. When the index v is arbitrary, the Lommel functions can be expressed in terms of the incomplete Weber integrals Qv (x, z) and 0, (x, 2). In this sense the function8 U, (w, z) and Y, (w, z) belong to a class closely related to the class of ICF’s. Our scheme gives a clear picture of the relationships between the different ICF classes and memebers and some more familiar functions. The ICF classes and their members are of great interest in both pure and applied mathematics (see e.g. Ill, part II). The series and asymptotic expansions obtained in Ill sometimes enable concrete problems to be analyzed and give results in a closed analytic form. In cases where these series cannot be applied in practice, tables for computing the ICF’s are needed for operational work. It has only been possible to evaluate the ICF’s in very special cases by means of

M. M. Agrest

46

the existing tables. series of ICF tables. in Table

Work was therefore started The main characteristics

1. The tables

aspects

concerning

conventionally

touch on some aspects

modified

incomplete

complete

series

of tables

The incomplete

integrals

the various

Modified

14 were published of tables

in L41. Here

No. 60 for the

and the method for using

the

ICF’s.

incomplete

Lipschitz-Hankel

Lipschitz-Hankel

in [31. Some

No. 16 are discussed

of the compilation

Lipschitz-Hankel

to evaluate

2.

numbered

the compilation

we shall

on the compilation of a special of these series are summarized

integrals

integrals

(1.6) of the modified

Bessel

and

MacDonald functions can be evaluated in the case of real arguments (a, z) either by means of the tables given in [21 for the Lommel functions if Ial > 1, or by means of the tables given in [3l when Ial < 1 (see [ll, pp. 315-316, or [31). Until recently there have been no tables when a = io is purely imaginary. We observe

as in 141 that the following

proved (other recurrence (1 -

$)le,+,

relations

(a, 2) =

-

for evaluating

recurrence

for integrals

(2~ + I)lev(a,

the integrals

relations

(1.6) are given 2) -

(1.6)

are easily in [5, 61):

~-azZv+i[~~(~)

4

alv+l(z)

1, (2.1)

(1

a”)

-

whence, question

Kev+l

(a,

2)

=

(2~ + l)Kev(a, 2) 2Vl?(Y + l)a

when v = n is an integer, for the case v = 0. Let

e-aZzV+i[Kv(Z)

we only need to tabulate

Zco(a,p>= {

co.5

-

U&+l

(Z)]

the functions

in

-

(2.2)

(d) 20 (t) dt,

0

Zs,(o* P) =

s

(2.3)

sin (at) Z. (t) dt,

0

where,

as before,

Z,(t)

The first of these variable

g.

Hence

The compiled

= Z,(t), K,(t). functions

is even,

and the second

we only need to tabulate tables

No. 60 contain

odd with respect

to the

for u > 0.

the values

of functions

(2.2) and (2.3)

Evaluation

of incomplete

cylindrical

47

functions

P

0.5 FIG. 1.

The surface

u(a, p)=

e-p

J0sin(ut)J(t)dt

in the following ranges of the variables: u = O(O.1) 10, f o(o.05). P=

10,

0 (0.05) F,

~~2,s;

(2.4)

ts>2.5.

To facilitate the use of the tables, the functions Ic,(o, p) and Is,(a, evaulated with the weighting factor exp (-p). Figures 1-4 give a graphical representation

p) are

of functions (2.2) and (2.3).

By using the asymptotic expansion for the ICF in the Poisson form and (1.81, asymptotic expansions may be obtained for the functions (2.2) and (2.3) when

M. M.

e-PfcO(a,

Agrest

p)

FIG. 2. The surface

~(a, p)-

s-O* cos(af)lo(r)dl 0

&I (6,P) =

2 $&: -

+ 1) -t- PKO (P) c”s (w)

iK;_@$ [cm (ap)

$01 (a, PI +

vo2 (Q, P)

(2.‘) sin (OP)l -

- pKo (P) [qlrz (G, p) cm (CP) KS0 (Q, pf =

If(f

J @2) + PKO (P) sin (GP) +

%z

(6

PI

sin(ap)l, (2.8)

Eualuation

of incomplete

cylindrical

functions

49

In the above expresssions,

xF(--k,kfll,

as/(Go-I- 1)) +

- n; %;

0 I(pa)-2N-l] ;

2ak+lI? (k + 8/2) I’ (k + “Is - n) r P/a - 4 (P 1/(1 + wk+l

x F (- k. k + “12- n; 3/*a;qp

y ’

+ f>>+ 0 I(p+ZN--flI!

n = 0, 4. The range (2.4! of the variables fo, p) was chosen on the basis that, beyond its Kso (a,~)

a(-

P

FIG. 3.

The surface u(a, p) =

I

0

sin(&)&(t)&

50

M. M. Agrest

P

FIG. 4.

limits,

the values

tabulation

By using previously

of the functions

by means 3.

The surface

Tables

5 0

cos(at)&(t)dt

could be found with the chosen

of the first 6 terms of the asymptotic

Determination

published

u(a, p)=

of the numerical

14, 16 and 60 (see Table tables,

values

of

(2.5M2.8).

of the ICF’s

1) in conjunction

we can find the numerical

accuracy

expansions

values

with some of the ICF’s

E’(w, Z) and cn (10, z) and their memebrs, namely, the incomplete LipschitzHankel integrals and the incomplete Weber integrals, when their arguments are any combinations of real and purely imaginary quantities. In addition, the title of the series of tables enables us to find the values of the ICF’s E’(w, .z) when ul is complex provided complex and IwI = 1. Table

2 summarizes

that Re w = r/2,

the conventional

and the values

numbers

of tn (w, z) when w is

of the tables

for computing

the

ICF’s or their members in different ranges of their arguments. The conventional numbers 8, 12, 30, 31. 32, 34 and 52 denote respectively the previously published tables of i2. T-l 11.

Evaluation

of incomplete

cylindrical

51

functions

We shall here consider in turn those ICF’s which can be evaluated by means of Tables No. 16 and 60. The evaluation

of the ICF in the Poisson form

&*(a, p), En*(ia, p), &*(u, Q), E,*(ia, or similarly,

ip)

of the integrals a

a.

c

CO.3

;sin

(o

cos t) sinw

‘OS (psht)sh!=%8, so sin 4 exp(-psht) sh*tdt, s

t dt,

a c i

exp (-

p cos t) sin%* t dt,

0

II =

0, I,

was discussed in detail in 11, 31. The numerical values of all these functions may be found by means of tables already published. Tables 16 and 60 are only required for finding the values of the ICF’s E,+(n I 2 + ia, p), &+(n / 2 -jh, 6~). But it is easily shown that evaluation of these functions is identical with evaluation of the ICF in Bessel’s form en (a, p), ~,(a, ip). Hence we do not need to consider them separately. When the variables a, z have arbitrary real and imaginary parts, the incomplete Lipschitz-Hankel integrals Ze,(a, z) (see (1.6)), may be evaluated by means of the tables appropriately numbered in Table 2. The functions Ze,(a, p) and Ze,(a, ip) of interest here are tabulated directly in Tables No. 16 and 60. Computational expressions for other ranges of the variables (a, z) are given in 11, 31. The ICF’s in Bessel’s form r, (a, p) (see (1.21) can be evaluated by means of Table No. 16. For, if we use the reduction formula

Jv(emxQ-) Nv(.?%z)

=

pvni&(Z),

= e-mvrriNV(z) + 2i sin Mvn ctgvnJ,(z),

for the cylindrical functions 1121, it is easy to prove the following relationships for the incomplete Lipschitz-Hankel integrals: Je0(-0,

-p)

=c -Jeo(o,

fleo(--o,

-p)

= -Xeo(o,

P), p) -22iJeo(o,

(3.1) p).

52

M. M. Agrest

On now substituting

following expressions ~kI(o,

for E: (w, z) from (1.7) in (1.11) and using (3.11, we find the for the E,,(a, pf:

P) = i i? sh ‘dt = aJ, (p) - -& ffo (P) -~IJo(W+-w)- 1% (p) Jeo (-

nis, (a, p}= i@

sh f-f&

=

1 _

$

(3.2)

Q, P)) cb a,

(f-@P)

(P) -

+aJ1

(3.3)

0

+

-4+(P)+

cb o (fir1 (P) Jeo (-

6,

P) - JI (P) Neo (-

6, P)),

where we have put (I = sh a for brevity. Let us now show that, when w = a and z = ip, the r, (w, z) can be evaluated by means of Tables No. 60. For this, we replace p in these expressions by ip and use the following easily proved equations: Je, (a, iz) = ei*(vi*~~)le.,(ia, z),

Ne, (a, iz) = -

eiv~ley (ia, 2) -

-z

<

After simple algebra we get (I = chaIIo (p) s sin(psht)dt

J-iKe, Td

(ia, z),

argz < n/2.

KCO

(qp)

-

KO

(P) ICO

(6,

p)I -

+0(P)*

0 0

s 0

cos

(p sh t) dt = ch a [K,, (p) Isi, fcr, P) -

10 (P) fcso (5, P)I + of, (P),

a s

6' sin (p sh t) dt =

+arl(p)-

‘-ccgS(5p)

0 - ch a I RI (p) iso (a,

P) -!- 11 (P) KSO (6 P)I,

a c

.= 1 + “i!$@ -

e-‘cos(psht)dt

++

L1 (p) -

b -

cha [K,(p)

Ic, (0, P) t 11 (P) Kco (a, !-‘)I+

Here, as above, CJ= sh a. Since computing

they are quoted the functions a

sf 1

co9

o sin

in [l, 31, we shall

omit here the expressions

en (ia, p) and tn (ia, ip) or integrals of the form

psh t -

nt)dt,

i zy: (nt) exp (p sin t) dt,

for

Integrals

Weber

Incomp!ete

ICF

Bk?SSel

Integrals

Hankel

Lipschitz-

Incomplete

-i-

32

PI

i 8.

24

31

t0 ‘iz _zP,P

12,

AngerWeber functions

Incomplete

j

Nedix, p) 8, 31

Je,(ir,

8,

16

BeaaelStruve

functions

(a,

1

PI

PI

(

fP,P j

30, 34, 52

aI

~.(~P. P)

60

I a/*<

30

Keo(a,

fe0

14

functions

modified

hcomplete

Hankel

Incomplete

(ia, P)

60

&(x/Z+

(+&ia,

ia, ip

iP)

Argument

TABLE 2

60

_,(qP,Pj

W($P,P

14

modif ied Anger-Weber functions

Incomplete

60

Ke0 (ia,

P)

fe0 (ia, P)

34, 52

Kn (a, P)

(M,W

j

po

14

9)

P

P

p, p

(W,

ew (W t -P, 2

Qo (q

14

en( exp

fexpfi3).

F 0

(

14

W,

p, p )

ip)

exp (ia) P, P 2 > 14 -

Q. (q%i

Ww

(exP @a), ip)

54

M. M. Agrest

and also the expression

for computing the ICF’s sn (&Or, p)

and sn (+,

iP)

We now turn to a consideration of the incomplete Weber integrals, confining ourselves as before to the cases in which Tables No. 16 and 20 are applicable. We shall quote here the expressions for computing the integrals of this type in the form in which they are most commonly encountered in applications. After some simple algebra we obtain from (1.12): P

s

IO (f/u) cos (hu)

au=

0

+

sin !y (10(VP)

cos(l/~)

(QJw - 1 +

{Blc0 (a, VP) --l~(l/P)

2h

+ cos (1/4h) 2h

c0s

vso(a, VP)1 +

sin (a VP)),

(1 - BISO(a, VP) - 10 (r/P) cos (a VP))*

where a = 1/41Ll/P - LIP,

B = 1/4hl/PfUP.

(3.4)

We can similarly obtain from (1.13): cd

s

Ko (1/u) cos (hu)du= -

sin ;y

@KS0ta,1/P)+ Ko (1/P)cos(a VP)-I-

P

+

ln PAV/p))+

{Ko

( 1/p)sin (01VP)t+

-

WC0

(a,T/P+ ~/2)~,

00

s

Ko (vu) sin (hu) du =

cos g4&’

(Ko (l/P) ms (a VP) + BKso (a, VP) +

P

(K. (T/p) co.5(a i/p)

t ni2 -

WC0

64

I/p+@?),

where a and J3 are given by (3.4) as before. The corresponding incomplete Weber integrals Q,Cx, p) and P,(x, p) may be evaluated by means of Table No. 16 in accordance with the expressions P Jo (VU) +U& = exp (--$f4h) (1 + &rPo (- a, 1/p) - JO (lip) @‘P}, s 0

Evaluation

Expressions remaining

for evaluating cases

the incomplete

functions

Weber integrals

and B. B. Tavdidishvili

No. 16 and 60 (both to be published

under the titel nepol’nykh

cylindrical

55

in question

in the

may be found in 11, 31.

M. M. Rikenglaz on Tables

of incomplete

“Tables

integralov

of the Incomplete

assisted

as co-authors

during 1970 by the VTs AN SSSR

Lipschitz-Hankel

Lipshitsa-Khankelya)).

in the work

Integrals”

(Tablitsy

The graphs were drawn by

N. G. Furs. Translated

by D. E. Brown

REFERENCES

1.

AGREST, M. M. and MAKSIMOV, M. Z. Theory of Incomplete Cylindrical and Their Applications (Teoriya nepolnykh tsilindricheskikh funktsii prilozheniya), Atomizdat, Moscow, 1965.

2.

BARK, L. S. and KUZNETSOV, P. I. Tables of Cylindrical Imaginary Variables (Tablitsy tsilindricheskikh funktsii peremennykh), AN SSSR, Moscow, 1962.

3.

AGREST, M. M., BEKAURI, I. N., MAKSIMOV, M. Z., ORLOVA, L. A., RIKENGLAZ, M. M., KHAIKHYAN, N. V. and CHACHIBAYA, TS. SH. Tables of Incomplete Cylindrical Functions (Tablitsy nepolnykh tsilindricheskikh funktsii), VTs AN SSSR, Moscow, 1966.

4.

AGREST, M. M. and RIKENGLAZ, M. M. The incomplete Zh. uychisl. Mat. mat. Fiz., 7, 6. 1370-1374. 1967.

5.

LUKE,

6.

WAI-KWOK NG, E. Recursive Bessel functions, J. Math.

7.

SCHWARZ. L. Untersuchung einige mit den Zylinderfunktionen nullter Ordnung verwandter Funltionen. Luftftihrforschung, 20, 341-372, 1944.

8.

STEEL, W. H. and WARD, Y. Incomplete Bessel Cambridge Phil. Sot., 52, 431-441. 1956.

9.

DEKANOSIDZE, E. N. Moscow, 1956.

10.

KARPOV, K. A. (Ed.). Tables of Generalized Integral obobshchennykh integral’nykh sinusov i kosinusov),

11.

WII,KES, M. V. A table of Chapmans grazing Sot., 67B, 304-308, 1954.

12.

GRADSHTEIN, I. S. and RYZHIK, Products (Tablitsy integralov, 1964.

Y. L.

Integrals

of Bessel

Tables

Functions,

Functions of Two ot duukh mnimykh

Lipschitz-Hankel

McGraw-Hill,

and Struve functions,

Functions

incidence

integrals,

N. Y., 1962.

formulas for the computation of certain Phys., 46, 2, 223-224, 1967.

of Cylindrical

Functions i ikh

integrals

of

Proc.

of Two Variables.

AN SSSR,

Sines and Cosines (Tablitsy VTs AN SSSR, Moscow, l!X6. integral

Ch(r,

I. M. Tables of Integrals, Sums, Series summ, ryadov i proizvedenil), Fizmatgiz,

Proc.

Phys.

and Mosvow,