Common Zeros of Two Bessel Functions Part II. Approximations … · 2018-11-16 · MATHEMATICS OF COMPUTATION VOLUME 41. NUMBER 163 JULY 1983, PAGES 203-217 Common Zeros of Two Bessel

MATHEMATICS OF COMPUTATIONVOLUME 41. NUMBER 163JULY 1983, PAGES 203-217

Common Zeros of Two Bessel Functions

Part II. Approximations and Tables

By T. C. Benton*

Abstract. In [ 1 ] it was shown that two Bessel functions J„(x), J (x) could have two zeros which

were common to both functions, and a computer program was made which takes approximate

values of v, ¡i and _/„ A. —Lk anàjvk + n =7íi,/1 + m and from them computes the exact values.

Here it will be shown how to find the necessary approximate values to initiate the computa-

tion. A table of the smaller ratios m : n with the orders of the functions less than one hundred

is given.

1. Introduction. In our paper [1] on common zeros of two Bessel functions we

developed a computer program which takes rough approximations to the orders p, v

iv > p) of the Bessel functions of the first kind Jrix) and Jßix) and to the positions

of the two common zeros and derives the exact values. In this paper methods of

obtaining the initial approximations will be discussed and a table of values given.

2. Notation. It is understood that the orders p, v are real numbers > 0. The zeros

of J„ix) will be denoted byj„ s, andjv, is the first positive zero withy, J+1 >jv s, s is

referred to as the rank of the zero. If the functions /„(x), Jyfx) have a pair of

common zeros jvk =jll<h and j„k+n =j^h+m, it will be convenient to speak of n

intervals of Ji[x) covering m intervals of JJ^x), it being understood that the intervals

are the segments between two successive zeros and the numbers h,k,m,n are all

integers.

3. Properties of the Zeros. The sequence of zeros of /„(*), call it [jv J, is an

increasing sequence. If 0 < v < {-, the intervals y„J+1 — jvs are less than it and

approach it ass -> oo, but for v > { they are greater than it and decrease towardtt as

î -» oo ; for v = { they are all exactly m in length. If s is fixed andy„ s is considered as

a function of the order v, j is a continuous increasing function of v with

(1) -ff = 2yrt,jf tfo^sinhOe-^'rf..

Proofs of these facts are contained in Watson's treatise [2], particularly in Chapter

15. Some other properties are contained in the following theorems.

Theorem 1. Ifs, t are fixed integers, t > s, andjv s,jv, are zeros ofJjix) where v is

real and > 0, then jv t — jv s is an increasing continuous function of v.

Received September 2, 1982.

1980 Mathematics Subject Classification. Primary 33A40, 33-04.

*Emeritus Professor of Mathematics, The Pennsylvania State University.

©1983 American Mathematical Society

0025-5718/83/0000-1362/S03.75

203

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

204 T. C. BENTON

(2)

Proof. djv s/dv is given by (1) and

^ = 2jPJfU2JvJsin\i<?)e-2"U<p.

Now y, s andy„,, are not functions of the variable of integration t>, so if y„ r sinh <p

j„ s sinh 6 andy„, cosh 4>d<p — jr s cosh 8 dd, (2) becomes

d¿,.

'o^ = 2¿ lfCCK0i2j^ssinhe)e-2-3KSinhaj^/j'-')únhe)

Jv ,s

j»,t

[=^sinh0-1/2

cosh 6 dd.

Since^//^, < 1,

./,.2i»arcsinh( — sinh0 < 2parcsinh(sinh0) = 2vd,

Jv.l I

whence

-2i»arcsinh(0,,,/j, ,)sinh9) -> p-2v8

Jv,1 + I ̂ sinh07r.i

1/2

<[1 + sinh20]l/2 = cosh0

So it follows that

^ > 2;,,J/o°C/ro(2;r,.ssinhc9)e-2^rfö = ^f,

and so 3(_/„, -jv¡s)/dv > 0 for all s and < with / > s. Q.E.D.

Theorem 2. If JJ(x) and Jîx) have two common positive zeros with n intervals of

Jj(x) covering m intervals of J^fx) and v > p and v > \, then m > n.

Proof. Let the common zeros be j„ k =jfLh andj„k+n = yM+m. By Theorem 1,

Jß h + m ~ Jn.h IS an increasing function of p. So if p is increased to v,jvh + m — jv h >

h.h+m ~h,h =L.k + n ~Jv,k- Since " > r*> L.k > h,k> but yM = jw%h > j^k therefore

h > k. Now if k is increased to h, jvh+n - j„h <jr,k+n - jn<k if v > { (Watson

[2, bottom of p. 515]). It follows thatjvh+m - jvh >jVth+n -jv<h and, since m, n are

integers, m > n.

If one makes a diagram plotting v as abscissa and jv s as ordinate, keeping s fixed,

a family of curves Cs is = 1,2,3,...) is obtained. As p -» oo the slopes of these

curves approach different values. The slopes of the chords connecting j]0,ooo.s ar*d

y 10 ooi j can be easuV computed and for s = 1, m— 1.00133; s = 4, m = 1.00385;

í = 50, m = 1.02125. This is an obvious consequence of Theorem 1, for in order that

the differences of the ordinates ofjvt andy,^ should increase the curves must diverge

as v increases. If the value of v is fixed and s increases, quite a different situation

occurs. From Watson [2, p. 508,15.6 (2)]:

"dfv,s _ 2v Ch.tr->, \dtV*,s z-v (J... T2( t\dt_

9" " i J2 i i Wo t '

As s -> oo,y„iS -h. oo, f^J2it)dt/t = 1/2»» (Watson [2, p. 405]). Now if 5 is large, jvs


COMMON ZEROS OF TWO BESSEL FUNCTIONS. PART II 205

can be made larger than v2, so that the first term of the asymptotic series gives the

good approximation

e = 2x^-^2j;y2cos{j^-v-f-l).

Thus j„ , = 7T-/2 + kir + vm/2 + tr/4, where A: is a large integer. So Jv+Xij„ s) =

2x/2ir-]>2j;Y2 cos km, and jp,sJ2+UJ - 2/m as j¥J - 00. Then dj^Jdv -*\/2,

and this is independent of the value of v. This diagram is of interest because finding

a pair of Bessel functions with two zeros in common is equivalent to finding a

rectangle with sides parallel to the axes having all four of its vertices lying on the

curves Cs. The diagram suggested the proof of Theorem 2.

4. Note on the Computation of Zeros of Bessel Functions. The Royal Society

Tables 7, Bessel Functions [3] gives a table of the zeros / of J¿x), 0 < v < 20.5

(0.5), 1 < s « 50 (1) together with the formulae for their calculation. Olver's

uniform asymptotic series [3, p. XIX and Section 8, p. XXXVI] is very powerful for

the calculation of zeros when v > 20 or s > 50. Using a TI59 calculator, a program

was constructed to avoid the complicated use of Everett's interpolation formulae

suggested in [3]. This was replaced by Newton's method for solving the equation

t/z2 — 1 — arceos l/z — c = 0. The coefficientp, was computed so that, iny„ s = vz

+ px/v — p2/v3 + ■ ■ ■, the first two terms were sufficient in most cases to give at

least five places of decimals, which is enough for the needed approximations.

5. The Difference Function Div). Consider the expression ij„ik+n — jvJc) —

ijp h+m ~Jn,h) withy„j;. =jli_h, where k, h, n and m are fixed integers. Since k and h

are fixed, the value of p will be a function of v and for a given v only one value of p

exists so thaty„ k =jlí¡h- If there were two different values, say p,, p2 > p, such that

j h =fn h' this would contradict the fact that^ h is an increasing function of p. This

makes the given expression (with the extra equality) a function of v alone. Call it

Div). Since by Theorem 1 each parenthesis is a continuous increasing function of its

order iv or p) and since the difference of two continuous functions is continuous, it

follows that Div) is a continuous function of v. By a well-known property of

continuous functions, if values vx and v2 exist so that Divx) < 0 and Div2) > 0, there

must be a value v such that v is between vx and v2 and Div) — 0. Now there will be a

value p such that j-k =j-A, and because Div) = 0, it will be true that 7- k+n =j^h+m

and the two Bessel Functions J£x) and J^fx) will have two zeros in common.

6. Some Approximations, (a) Consider the case m = 2, n = 1, k — 1 so that

U,2> ^,1] must cover tX^+2, j,ih]. Now two intervals for p small will have;„,,+2 - jvh

approximately equal to 2m. From formulae developed by Olver [3, p. XVIII]

la -fv,\ = 1.388505k1/3 + 2.125093i'-|/3 - 0.79333»/-' - 0.75291f-5/3 + • • ■ .

If three terms are kept this equation becomes

1.389x4 - 6.283x3 + 2.125x2 - 0.079 = 0, x = vx/\

This has a root x = 4.156, and so v = 71.78.

Suppose that v = 72 is used as an approximation,y72 , = 79.96646. From tables [3]

J0ix) hasy025 = 77.75603 as its largest zero not greater than/^ , andy, 25 = 79.32049

<jnx = 79.966464 <;', 525 = 80.09813. By calculatingyM25 with different values of


206 T. C. BENTON

p it was found that/ 4i5209025 = 79.966464 — j12,. Also

j122 = 86.255443,

j12 , = 79.966464 ,

71.4.5209.27 = 86.250448,

7i .415209,25 = 79.966464,

6.288979 6.283984

£(72) = (y72.2 -772.,) - U.27 ~fß,2s) = 6.288979 - 6.283984 = +0.004995.

Now use 71 for the approximation.

771.2 = 85.196177, y075076<27

771.1 = 78.931722, 7<>.75ü76,25

6.264455

D{11) = 6.264455 - 6.283332 =

= 85.215054,

= 78.931722,

6.283332

-0.018887.

These calculations show that v lies between 71 and 72, p lies between 0.5 and 1.5 and

the zeros are between 78 and 80 and 85 and 87. These approximations in the

computer program gave the results v = 71.87224, p = 1.33143,/,, =jfL25 — 79.83629,

7,.2=7,,27 = 86.12017.

(b) In order to find more pairs of such Bessel functions, suppose that the interval

[7n,25' 7,1.27] is replaced by [y^, 7^,2«]- It will be shown that this does not lead to

values such that Divx) and Div2) have opposite signs. Taking p = 0, j02t =

80.897556. If/o, =y026, vQ = 72.900085. Then

]vn2 - 87.208432, y028 = 87.180630,

¿eJ = 80.897556 , Jo26 = 80.897556 ,

6.310876 6.283074

D(v0)= +0.027802.

Similar calculations yield /J(74) = 0.05410. It becomes impossible to find a negative

value. Hence to obtain further values, the ranks of the zeros of Jîx) must be

decreased so that [,/24, yM26] is the next interval to be considered. In this case

similar methods lead to v = 72.06767, p = 3.50186,/ , = /, 24 = 80.03849 and/ 2 =

j^ = 86.81724.(c) This decreasing of the rank of the zeros of Jîx) leads to [7^4, /, 2] and stops

there since [y^3, j ,] must have^ , =/ ,, which implies p = v contrary to Theorem

2. However if the case [./-10, y'12] is calculated, then 655 < v < 657, 582 < p < 585,

and the zeros lie between 671 and 674 and 683 and 686. (Since Jl00ix) is the function

of largest order for which the computation of Bessel functions is validated, it is not

possible to carry out this determination by the computer program.) The next case

[/i,9> 7^11], however, is quite different. In this case Div) < 0 for all v. The following

table was found:

v

Div)

800-0.357

1,000-0.335

2,000-0.281

6,000-0.231

20,000-0.214

100,000-0.248

400,000-0.335

1,000,000-0.432


COMMON ZEROS OF TWO BESSEL FUNCTIONS PART II 207

Hence decreasing the rank fails to yield new pairs before the extreme limit is

reached.

7. General Method of Approximation. Suppose n intervals of •/„(■*) cover m

intervals of J Ax) and/ k is the smallest zero of •/„(*) belonging to the n intervals.

The m intervals of Jîx) will have/, h + m — / h very nearly equal to mm if p is small.

Determine v so that/,it+n - jvk < mm but/+1 k + n -jr+1%k > mm. This fixes jvk.

Now, from tables ofj0s in [3], find the nearest jQh tojvk butj0 h <j„k. This fixes h.

From the sequence of values ofj0h,j[/2h,jXh, etc. in the tables [3] find the values j h

andyM+1/2A nearest to jvk. Then by interpolating and calculating zeros p can be

determined so that/, h =jpk to a close approximation. From these results/ k+n and

Jv.h+m can be found and thus Div) can be calculated. If p > l/2,jfih+m — /,,/, > mm

and/ k+n —jPtk was fixed so as to be < mm. In this case Div) will always be < 0. If

v is increased, keeping m, n, h, k the same, it may be that Div + 1) > 0, and a pair

of functions with two common zeros will be determined. If Div + 1) < 0, try

Div + 2) etc. As example (c) in Section 5 shows there are cases where Div) stays

< 0 no matter how large v becomes, in which case no pair of functions is found. If

p < t it may result that Div) > 0. In this case changing v to v — 1 (not changing any

of m, n, h, k) may give Div — I) < 0 and determine a pair of functions with two

common zeros. As example 6(b) shows Div) can remain > 0 for all v, and in this

event no pair of suitable functions will be found.

Suppose that the first case mentioned in the previous paragraph occurs. Then

Jv.k+n -Jv.k < mm but Jv+\,k+n -Jy+\,k > m7r- Now change the rank h of jßh to

h + 1. Then there will be a new value v' so that/. ^ = Lh+\. Since/, A+l — j h > m

in > y) and from the properties of the curves Cs discussed in Section 3, 1 <j„+i k~

)v.k < t/2. So it follows that v' > v + 2. Then j„,k+n -/, k >jy+hk+„ -/+.,* >

m7r.AlsojM+1+„, -jlltk+x <jlltk+m - j^ since p> \. Then Div') > O.Ewenif ¡j. ̂ {,

Jp.,h+m+\ ~J]x,h+\ < mm and Div') > 0. Notice that increasing v makes the first

parentheses of Div) increase, while the second parentheses cannot increase beyond

mm, so that Div) is always positive. Therefore most cases of suitable pairs of Bessel

functions can be obtained only by decreasing the rank h.

If h is decreased, new cases may result until the case h = k + 1 is reached. The

process stops here because h = k would require/ A. =Lk, and this implies p = v,

which is not true by Theorem 2.

8. Comments on the Approximations. In order to obtain one set of approximations

for determining a pair of Bessel functions Jv(x) and J/fx) with two zeros in common,

at least eight zeros of the functions are needed, four to find a place where

iJv.k+n ~Jv,k) - ijfi.g+m -7^,g)<0 and four where this difference is > 0. The

uniform asymptotic formulae of F. W. J. Olver enables one to calculate the zeros

very easily and without them this work could never have been carried out.

A limitation on all the calculations here is imposed by the fact that our large

computer program for calculation of Bessel functions is only guaranteed to produce

correct results for orders not exceeding 100. In order to determine how many cases

arise, it is necessary to have a table of zeros of 7100(x). This has been computed

using three terms of Olver's series, so the values are correct to 8 places of decimals at

least.


208 T. C. BENTON

S 7ioo,î S 7ioo,j1. 108.83616 58968 14. 169.89299 68759

2. 115.73935 12229 15. 173.75627 22487

3. 121.57533 10257 16. 177.57743 71980

4. 126.87075 61516 17. 181.36076 93720

5. 131.82393 46667 18. 185.1099146998

6. 136.5357182239 19. 188.82800 88671

7. 141.06584 76591 20. 192.51777 03132

8. 145.45320 90903 21. 196.18157 27018

9. 149.72480 08232 22. 199.82150 22255

10. 153.90027 12271 23. 203.43940 27048

11. 157.99444 31441 24. 207.0369127349

12. 162.01882 07206 25. 210.61549 54381

13. 165.98245 03531 26. 214.17646 28640

Consider the case of three intervals covering four 477 = 12.56637 andjx00 n — /100,8

= 12.54123. To cover four intervals with any Jßix) with p. > \ it will be necessary to

take v > 100. But jxo0 10 —jXOOtl = 12.83442 and this will cover four intervals. So all

cases from/,4,/ , up to/ 10,/ 7 must be considered. By this method it is possible to

find limits for any given m and n keeping v < 100.

9. Other Problems. Perhaps the most interesting problem raised by this work is the

question as to whether or not two Bessel functions can have more than two zeros in

common. If a pair with three common zeros exists, they will of course have three

cases of having pairs in common, so that if a complete list of pairs could be made,

there would be three cases where for the same pair of values of v and p there would

be entries in the list. This suggests that the list be arranged so that the order v is in

ascending order, and then that one looks at the values of p to see if any pairs are the

same. This was done with the values in the table and no pairs were found. Of course

this suggests that it is not possible for two Bessel functions of the first kind to have

more than two zeros in common, but it is also possible that the covering ratio m to n

has not been extended far enough to make such a case occur. So it seems that there

is not enough evidence to make a conjecture at this time.

Another question is whether there could be a third Jxix) which also has the same

two zeros as J^x) and /„(x). If jrk = /,ig and/, g is large enough, it is easy to see that

A < p can be determined so thatyx , =Lg, but can it be found so that the second

common zero is also a zero of/x(x)?

Another interesting question is the number of cases of pairs of Bessel functions

with common pairs of zeros. It is obvious that the method outlined in this paper will

produce at most a countable set of such functions. However, it is not proved that

this method of obtaining the pairs of functions necessarily produces all of them.

10. Acknowledgments. Thanks for support from the Mathematics Department and

the Computation Center of the Pennsylvania State University are gratefully given.

Special thanks are due to H. D. Knoble for assistance in working the computer and

for preparing programs and arranging the tables.



11. Tables of Those Bessel Functions Which Have Two Zeros in Common. In these

tables u, v correspond to v, p in the above explanation, and a, b, c, d are used for the

ranks of zeros. The calculations of the exact values were done by the large computer

and entered on punch cards, which were then used to print the table. This avoids the

difficulties of copying and proofreading.

* b c d u v Ju,« " Jv.b Ju.c - Jv.d

1 0VW 21 25, 2 271 24, 2 26I 23, 2 251 22, 2 241 21, 2 231 20, 2 221 19, 2 211 18, 2 201 17, 2 191 16, 2 181 15, 2 17

2 over 31 12, 3 151 11, 3 141 10, 3 131 9, 3 121 8, 3 111 7, 3 101 6, 3 9

2 20, 4 232 19, 4 222 18, 4 212 17, 4 202 16, 4 192 15, 4 182 14, 4 172 13, 4 162 12, 4 152 11, 4 14

3 28, 5 313 27, 5 303 26, 5 293 25, 5 283 24, 5 273 23, 5 263 22, 5 253 21, 5 243 20, 5 233 19, 5 223 18, 5 21

4 37, 6 404 36, 6 394 35, 6 384 34, 6 374 33, 6 364 32, 6 354 31, 6 344 30, 6 334 29, 6 324 23, 6 31

71.8722363572.0676723972.4745464573.1464978474.1565675275.6067287077.6433865880.4834213784.4713336890.1366015098.51353133

30.9923961131.2031320631.9340594833.5572869636.8964030944.1796024863.97506249

51.0130653151.2516644851.7837010752.7144733134.2052848756.5157651460.0907906565.7575635275.2356354592.72888373

70.5137281070.7312673271.1472825971.8090533672.7798638474.1463167276.0300181178.6069881282.1414268387.0478030094.01306654

89.7620801989.8320130790.0291395990.3749065590.8957779091.6247776992.6036350193.8858219795.5409474397.66126296

1.331432713.501864955.856813728.44339038

11.3260027414.5949747618.3808172522.8789415428.3948309433.4322306244.38014463

0.148509832.336925114.991084178.44729585

13.4566895322.0755988842.33739479

1.182741313.395997695.871762988.70487222

12.0412707116.1171307221.3396558528.4714758039.1112656957.20607243

1.809478144.003885506.373775918.96765542

11.8365930015.0609526918.7511904423.0678626528.2530896034.6386802343.01111475

0.230489952.292976004.469129056.778170469.24386741

11.8959461614.7720619917.9205973021.404 7205325.30841526

79.8362918780.0384907680.4594064281.1544366282.1989422783.6980182385.8025685038.7369583192.8491940193.70813596

1C7.30791651

37.1500592037.3732391638.1469236939.8629533143.3344389151.0319764971.65529954

63.3953112064.1513346264.7220447365.7198884467.3166174969.7877418873.6034365579.6341343139.67940950

108.11122386

90.0047179790.2394432490.6382345391.4019019692.4483158093.9201409795.9471619598.71669945

102.50902107107.76224715115.19967326

115.81642269115.89213060116.10549232116.47968542117.04325269117.83175844118.89006332120.27554382122.06273780124.33020817

86.1201729186.3271440836.7579700487.4692937288.5381221590.0717703592.2242077095.2241518699.42600032

105.40839993114.13127176

46.5742169546.3120730047.6362595449.4622934053.2014333161.2890825132.93354995

73.3212446273.5373920974.1305380475.2174777876.8759923279.4411087382.3983070489.64415388

100.02751939119.02544055

99.4310885199.67249383

100.13404423100.36791936101.94380021103.45675044105.53980814108.38431713112.27351981117.66363311125.29334253

125.24114670125.31353367125.53665326125.91917724126.49526339127.30122683123.26285770129.79869564131.62475296133.96143390


210 T. C. BENTON

•lu,a " Jv.b Ju,c " Jv,t

OVER 4

1 33, 3l 32,1 31,1 30,

28,29,

373635343233

3 OVER 41 8, 4 12

4 114 10

4 94 3

18171615141312

232221201918171615

2928272625242322212019

8 348 338 328 318 308 298 288 278 268 258 248 23

95.2368292295-3375822995.5768839195.9790093097.4002375596.57413650

22.0719142823.0912086725.8476407333.7470532272.03933686

33.8875351034.2819559435.2024110836.9793856540.2840271746.7041373761.02535337

45.2905356345.8517303346.6481300047.9810432950.1100196953.4895117158.9895006868.4949626286.91853232

56.5857361256.7747277857.1899657857.8961110358.9845598360.5886638562.9105694266.2713359771.2116850278.7138351090.75301477

67.7979417168.0131004668.4190029369.0599770269.9949826171.3041188973.0989068175.5391502978.8618180583.4331657589.8489745299.14324407

0.669368582.757249314.966009937.31691631

12.558139859.¿3647593

2.149754545.078574069.60051267

18.8948476057.02319266

1.387913843.746603216.58451988

10.2053718415.2301211623.1403600038.45011276

0.297875194.808075917.53268192

10.7468987814.6899305319.7828736126.8369787737.6231599456.79815250

1.127258873.299021715.676513928.31880635

11.3096513314.7716710418.8914756523.9667896630.5015673739.4165232152.57537773

1.889751064.085288076.454254449.03716694

11.8879811315.0802523518-7166961322.9450010327.9850077034.1779340542.0814360852.66839439

103.93764840104.04130932104.28750950104.70120212106.15306230105.31339078

27.6451202028.7378853531.6334476040.0633528780.00917544

45.3585538145.7898636046.7954230848.7329723752.3242338559.2626604374.59949930

62.515645S363.1346118564.0122782465.4793764167.8181370471.5199393477.5192309587.82489305

107.62399543

79.5186946379.7282264780.1884525280.9706523282.1752914283.9433271786.5101157490.2089379395.62811359

103.82060796116.8S880215

96.4135723896.6528919597.1042525497.8166896198.35524429

100.30800457102.29719224104.99727514108.66586750113.69925591120.73869352130.89060800

115.5041217:116.6114389»116.3664823.117.2949214¿118.8086031:117.9283659:

40.2262225:41.4410627;

44.6785707:53.8197856:96.5340131!

57.9289324:58.3887426:59.4602296!61.522S214.55.3392852.72.6909168»83.8562536:

75.0818006-75.7311323:76.6515983!78.1895774.80.6396657»84.5136752/

90.7823396!101.5256443!122.0915511:

92.0859409Í92.3036333!92.7819064:93.5945335!94.3458384:96.6863653:99.3455890:

103.1817551-

108.7963675!117.2745321;130.7730744!

108.9819291:109.2290566»109.6951161!110.4306331;111.502SC0SC113.00220S9C115-0546969»117.8396368»121.6218207:

126.8077227!134.0544817C144.4940644:


COMMON ZEROS OF TWO BESSEL FUNCTIONS PART II 211

Ju,a = Jv,b u,c v,d

OVER 46 36,

33,34,33,32,31,30,29,28,27,26,25,

(conciBucd)403938373635343332313029

41,40,

39,38,37,36,35,34,

7 33,7 32,

10 4510 4410 4310 4210 4110 4010 3910 3810 3710 36

OVER 51 20,1 19,1 18,1 17,1 16,1 15,1 14,1 13,

4 254 244 23

2221201918

1 12, 4 17

34,33,32,31,30,29,28,27,26,23,

39383736353433323130

78.78.79.79.80.81.82.83.85.38.91.95.

368304779608205519218234587982831804054701046974131253346126219754835232067177733505055566198024

90.91.

89.9742392290.0919639390.33404242

.72042528

.2754994592.0293815993.0196382094.2939995595.9133541097.95731019

59.2723981659.6996805260.4453733761.6173658263.3762137265.9726591369.8201186275.6491974184.87553135

91.0910524791.1634939691.3681407591.7279066392.2712190693.0337662494.0609477795.4113809897.1620302199.41590125

0.528205552.612273064.822530877.182304779.72098393

12.4761055315.4963542318.8459760422.6114444026.9118278131.9154706337.36791887

1.202019043.308992315.528986297.88013594

10.3848573813.0705435415.9717359019.1321218422.6079007426.47250147

3.052422965.430026588.09027636

11.1292026314.6910725819.0024441024.4392261331.6705465942.00366140

0.229733472.293769414.474691456.792824109.27344172

11.9483557714.8531481918.0553771621.6092825525.61286980

113.14151254113.24466801113.50260545113.94376493114.60383362115.52813861116.77539277118.42231074120.57173903123.26452562126.99804131131.75821505

129.90342884130.03491939130.30527130130.73669141131.35626949132.19739282133.30156801134.72159293136.52433526138.79713934

66.7732897467.2167653767.9914425969.2085290271.0340288173.7257283377.7124362933.7419102993.26598465

106.39054339106.46666913106.68171407107.05972393107.63050236108.43142612109.50998846110.92743322112.76406367115.12719467

125.70789597125.81393841126.07908374126.33236518127.21101689123.16102377129.44267152131.13473049133.34274957126.21066937139.94067036144.82504757

142.47020503142.60491550142.38133623143.32385073143.95854136144.8201222414S.95115314147.40531301149.25123014151.57813965

32.4941937732.9660541083.7900530983.0839632087.0232307839.8806378094.10374469

100.47897589110.52006732

122-09338743122.17755373122.40158225122.79517340123.33941721124.22314377Í25.34565343126.82046440123.73078101131.18767662


212 T. C. BENTON

a S 3u,a ° jv,b u.c Jv,d

4 OVER 51 7, 5 12

5 U5 103 9

17161514131211

7 217 20

191817161514

3 258 248 238 228 218 208 193 188 17

29282726232423222120

10 3410 3310 3210 3110 3010 2910 2810 2710 2610 2510 2410 23

18.0945716018.9719951721.8154322732.08334843

26.7661112527.0303941627.8679900529.7333263433.7131732043.1513692374.84710729

35.0527675435.3508291936.0727440737.4448228939.8848584544.2536896652.6317905371.51525742

43.1360837443.4337777144.0617240045.1523949046.9184403649.7230510454.2387778661.8561038475.94791897

51.1124099351.4006701851.9572664252.8682710754.2602397756.3268290759.3792955463.9499012171.0233864982.64740770

38.9291203059.0254352159.3022974659.3041553560.5910939961.7462360763.3881237663.6907920068.9209838673.3075452030.1840115190.31008383

1.393965894.203413438.33687798

20.44730831

0.385949872.629266445.401094159.12350938

14.8131975225.6287095557.65795798

1.061982543.336303456.00U39229.26632980

13.5211727719.3725541829.3857131649.15331390

1.565847943.839779616.417875549.42282453

13.0523077117.6454332023.3334048832.9291927048.14088554

1.978874294.244094736.756391369.59541396

12.8786257916.7863875821.6097252427.8485436536.4343801049.30614350

0.249305232.338102624.592816347.054662349.77920649

12.3435612016.3377981920.4843727325.4728994831.7263575239.9371184651.39505373

23.3591692424.3079992727.3698761438.30698562

37.5213083137.3140570738.7407984340.7991110045.1678020535.4288787389.26854371

51.1396615251.4734341452.2811233153.8135113756.5203582761.3709343970.5316014491.08510265

64.4390126564.824561S265.5318724766.7538569763.7416525671.8809683676.9132216985.34737296

100.30499286

77.6976436678.0239968378.6538025379.6836719581.2350171883.3831115487.0119096992.1256081899.99882373

112.33517232

90.7137455390.3231602491.1376111691.7073522292.6000854193.9091739995.7669417698.36713019

102.00476331107.15134752114.60785052125.34747537

39.0816937040.1378218443.6162996755.82262293

33.2288745553.5479094354.5572811356.7960298761.5343713572.60107539

108.65062753

66.3496409367.2061490163.0635357369.7038112172.5997549077.7497183837.51790632

109.14102367

30.2003196680.5543639081.3005206682.5944462184.6842429887.9900877593.23224645

102.13400137118.30553843

93.4095747493.7510625994.4099951795.4872664397.1304002799.56363791

103.14494382108.43077493116.63447490130.03249470

106.42155609106.53335901106.36240313107.45492327108.38321749109.74416631111.67496235114.37626377118.15325502123.49298:52131.22133834142.35443251



«bed_u_*_ju,a = Jv,b ju,c = jv,d

4 OVER 5 (coaeiauad)7 33, 11 387 32, 11 377 31, 11 367 30, 11 357 29, 11 347 28, 11 337 27, U 327 26, 11 317 25, 11 307 24, 11 297 23, 11 287 22, 11 277 21, 11 26

66.7914450366.8978944967.1636074667.6225368268.3188613169.3111064170.6784138372.5303792275.0230521378.3860770132.9711007839.3435263298.47006830

8 37, 12 423 36, 12 418 35, 12 403 34, 12 393 33, 12 388 32, 12 373 31, 12 368 30, 12 35S 29, 12 348 23, 12 338 27, 12 328 26, 12 318 25, 12 30

9 41, 13 469 40, 13 459 39, 13 449 38, 13 439 37, 13 429 36, 13 419 35, 13 409 34, 13 399 33, 13 389 32, 13 379 31, 13 369 30, 13 35

10 45, 14 5010 44, 14 4910 43, 14 4810 42, 14 4710 41, 14 4610 40, 14 4510 39, 14 4410 38, 14 4310 37, 14 4210 36, 14 41

74.6296034274.7427951374.9982751175.4227861476.0501112676.9235297578.0993693879.6522667781.6831696184.3313902487.7975153892.3730077698.50686880

32.4502958682.5681189182.8144203033.2107882683.7838580484.5668638485.6017961136.9424568638.6588780790.8438637593.6229385797.16994855

90.2579518790.3790360390.6171676690.9901231491.5194370092.2313794493.1583897894.3409424395.3301803497.69163689

0.564928442.662844514.907287417.329581899.97056531

12.8844523516.1447294219.8534282124.1562508429.2633041335.3201467943.4453707453.96029470

0.860042822.964157365.199177337.58977009

10.1671521912.9713889916.0547323119.4365736123.3609973527.3086612033.0161639339.2589767446.96031140

1.140457573.248830455.475399337.84007564

10.3674631213.0883119916.0415443019.2771233322.8602442726.8774661031.4461850436.72942338

1.409883603.521239385.740307653.08343183

10.5705050113.2258374316.0794746219.1689752022.5418865526.25925767

103.77421379103.89543164104.19795077104.72025713105.51227958106.53994270108.19206620110.29114612113.11081151116.90529720122.06174242129.19878622139-36692943

116.80233603116.93152410117.22294602117.70702683118.42203692119.41686165120.75492119122.51995410124.32473105127.82499678131.74113313136.89603947143.78130450

129.3072S026129.94190612130.22323908130.67599033131.33024103132.223664U133.40366017134.93077533136.88355324139.36570474142.51693531146.52990646

142.79433423142.93335906143.20573238143.63227212144.23733196145.05089312146.10949713147.45885493149.15649135151.27538050

119.48222085119.60771969119.92091236120.46161506121.28145319122.44855505124.05464913126.22614929129.14203455133.06432731138.39122333145.75845595156.24371347

132.51039776132.64331125132.94442254133.44374430134.18121438135.20718539136.53694969138.40666792140.7S235250143-87333449147.90763592153.21494020160.29929265

145.31568047145.65414703145.94355134146.40914239147.08198477148.00072742149.21404383150.73407396152.79141312155.34238002158.53014321162.70199433

153.30340057158.54553177158.92501205159.36261531159.93342604160.31799646161.90392294163.28797430165.02905052157.20224665


214 T. C. BENTON

u,a Jv,b

4 OVER 5 (continued)11 49,U 48,11 47,U 46,11 45,11 44,

15 5415 5315 5213 5113 5015 49

98.0356183198.1790431098.4099108598.7631146199.2563916099.91102993

1.670895933.784413285.996780613.32171632

10.7756020213.37811932

155.76911998155.91047040156.17483449156.57919927157.14375569157.89268496

"u,c Jv,d

171.47783067171.62256545171.89323546172.30728670172.38531983173.65209266

4 OVER 61 16,1 15,1 14,1 13,1 12,1 11,1 10,1 9,1 8,

2 25.2 24,2 23,2 22,2 21,2 20,2 19,2 18,2 17,2 16,2 13,

34,33,32,31.30,29,28,27,26,25,24,

555353555

66666666666

777777777 3277

222120191817161514

3130292327262324232221

4039383736353433

3130

44.9902733245.3189561246.0251266747.2760532449.3523173952.7613578958.5187630368.9730308590.92474163

66.4369280066.6568442067.0896145767.7898364468.8321376570.3211712872.4081911575.3198314879.4109110685.2631024893.93136303

86.6618760086.7723496787.0220037887.4357322038.0446290488.8880255890.0163866791.4954905393.4126566795.8862223299.08045886

1.541365793.335494196.467894919.58963019

13.4557558918.5302094425.7492663137.2985838159.58038671

1.666124463.862754156.249825088.37645522

11.8100014115.1452663319.0198297523.6407847329.3340238436.6414936246.53025575

0.842165362.940906165.164076737.53401062

10.0786658212.8334936615.8440980019.1701000822.8908924827.1144423931.99118885

51.8807710332.2247363752.9636609154.2717444356.4407858659.9969000065.9894675776.8353399399.49919211

80.3553436630.5393930931.0489101681.7921303882.8980476784.4768540986.6877143089.7684366394.09027982

100.26515836109.37415139

107.34948339107.46764678107.73465620108.17707808108.82805115109.72943223110.93433525112.51402692114.55943755117.19606391120.59704696

70.7337865071.1111346171.9169255273.3422302575.7022599679.5628700836.0464389197.72007262

121.91638427

99.2083860399.4562206799.94377716

100.73221402101.90485920103.57814264105.91964844109.17931329113.74641697120.26077083129.84918432

126.19935377125.32303565126.60249115127.06549741127.74667197123.63970423129.95050590131.60175186133.73966421136.49414539140.04492343

OVER 71 29,1 23,1 27.1 26,1 25,1 24,1 23,1 22,1 21,1 20,

363534333231302928

5 27

34.5103545884.6893262585.0348297885.5803594086.3688532237.4562364988.9166748890.8505769293.3971688896.75498075

1.6436499 73.801305006.105730918.58659867

11.2820546614.2419409517.5325371621.2441379925.5021147330.48633149

92.8394323393.0739830893.4302354593.9926794794.3054930495.9261920697.4309500999.42282032

102.04453883105.49940605

114.38310703115.07972144115.45922119116.05326099116.92373739118.11659709119.71746844121.83521616124.62034426123.28675274


COMMON ZEROS OF TWO BESSEL FUNCTIONS. PART II

•bed u v ju,a = jv,b

5 OF TWO BESSEL FUNCT

V

2.093816645.45160970

12.3672822747.98237712

3 OVER 61 6, 6 121 5, 6 1114, 6 101 3, 6 9

2 10, 7 162 9, 7 152 8, 7 142 7, 7 132 6, 7 12

3 14, 8 203 13, 3 193 12, 8 133 U, 8 173 10, 8 163 9. 8 153 8, 8 14

4 17, 9 234 16, 9 224 15, 9 214 14, 9 204 13, 9 194 12. 9 184 11, 9 174 10, 9 16

5 21, 10 275 20, 10 263 19, 10 25S 13, 10 245 17, 10 233 16, 10 225 15, 10 215 14, 10 203 13, 10 193 12, 10 18

6 25, 11 316 24, 11 306 23, 11 296 22, 11 286 21, 11 276 20, 11 266 19, 11 256 18, 11 246 17, 11 236 16, 11 22

16.1395628217.6137520122.8548124958.02932913

23.1203479123.8327742025.6257212129.8339973741.54204675

29.5913496929.9357424330.8370584332.6376001736.1291031043.2361129660.71283353

36.0863840936.3917810637.5853731739-3298087042.3121042947.5417684757.4800080979.92313004

42.2258895642.5102141643.1116109544.1540376945.3343795648.4843914652.7075483659.7209806672-3698045099.03135949

48.3712011348.5151673443.3872500149.5574216950.6270074152.2482214854.6602034458.2602565063.7375565172.54501731

1.833902844.496461338.16703856

14.1225712527.14617804

1.149173363.487737146.32581573

10.0027030415.2450508923.9135459042.45804181

2.499756414.969304617.89310552

11.5171889016.2997596323.1957134934.5485246957.84319171

1.526922423.790967956.349690319.31884090

12.8828538717.3536666423.3011969131.8808516845.8101460473.15077259

0.563520152.697191425.042736457.66536130

10.5597193514.1684023018.4161554623.7777326830.9250106541.18397591

21.2588639522.3383207628.4846310865.48084340

33.4647115734.2602534036.2561472140.9320146153.63759611

44.9901236245.4025923846.4214065948.4509162552.3441364660.2460297579.39317599

56.4952093357.0715967253.2031590460.1349221963.5594561769.4403527030.50980457

105.13740677

67.5711303667.8972870168.3366545269.7799554471.6993547474.7165522279.5020642787.39501745

101.48946187130.76869445

78.6391640678.8049922579.2334113080.0044530681.2334827383.0928314285.8515819789.9535720596.18451964

106.07300955

215

Ju,c = jv,d

40.1542380341.9841333248.4735953389.69121075

32.3310295153.2138175255.4252047860.5878030074.55831602

63.3431913354.2907152765.3955959967.5943592371.3043452380.31930574

100.81096220

75.3530545675.9744931077.1842253479.3014455332.9023054939.16620539

100.91839029126.91642482

86.4240458286.7694637487.4994390288.7626438190.7934820593.9834063399.03691912

107.35675256122.17084383152.79952637

97.4838030297.6631543598.1135623198.92406455

100.21566301102.16896723105.06553327109.36902393115.39853456126.24329534


216 T. C. BENTON

a b ju,a = K, Jv,d

5 OVER 6 (continuad)12 3412 3312 3212 3112 3012 2912 2812 2712 2612 2312 24

13 3813 3713 3613 3313 3413 3313 3213 3113 3013 2913 2813 27

14 4114 4014 3914 3814 3714 3614 3514 3414 3314 3214 3114 30

15 4515 4415 4315 4215 41IS 4015 3915 3815 3715 3615 3515 3413 33

5454535557586064687535

56740026.73982592.23167039.9331179103819188

.60770848

.84159964

.01951472

.59967679

.38517902

.91411822

60.6481470960.7655165361.0538317961.5523814662.3132972263.4072102864.9321565567.0281169069.9017988273.8710927779.4499433587.52348170

66.7870455966.9653869367.3097876967.8552901768.6473868569.7461876471.2327351673.2188696675.8631888479.3979517384.1767026990.76370747

72.8391484772.9349973673.1667898873.3594902274.1444812674.9617487676.0630151477.5163381179.4130338281.8784087085.0889664439.3011061594.90127200

1.649971453.856490246.266330818.93710543

11.9458234015.4061771819.4371871324.4524156530.7352851339.0985452651.00986134

0.631028922.739992405.007692707.47070894

10.1776659113.1945777716.6132305320.5652356625.2451192730.9532639338.1769429247.75364012

1.667361023.332930256.152714008.65935276

11.3952923214.4167245917.7995636621.6488151926.1137609031.4136270837.8831477246.05838656

.62407731,71305567.92825133.29289357.33622474

12.5955637415.6192671718.9710855422.7367346527.0241049832.0296696537.9659184045.20942649

89.7571929890.0141335490.5243133191.3566950592.6071404594.4128092396.97633627

100.61294439105.33153214113.51992345125.36197492

100.73604913100.37196962101.20577198101.78270031102.66257379103.92611350105.58435596108.09725326111.39599826115.93666427122.29013127131.43135148

111.77811207111.98502225112.38447396113.01686400113.93444584115.20601462116.92393603119.21509256122.25850222126.31436233131.77855575139.27471907

122.71644542122.82783470123.09715236123.53330836124.23247027125.18062454126.45703437128.13938092130.33145426133.17502164136.36876497141.69970419148.09781100

108.6C913931108.37776497109.41110780110.23116941111.53796192113.47445509116.15216404119.94733824125.38993141133.39951924145.71705241

119.33572098119.72719106120.07460992120.67503032121.59062920122.90525744124.73469136127.24327332130.57203037135.33910037141.98427852151.46392290

130.62930112130.34387303131.25810229131.91383914132.36520654134.1S3¿1339133.96401142133.33816878141.49080133145.69094325151.34513332159-09696791

141.56607700141.53124246141.93969341142.43127536143.13337933144.U347575145.43273165147.17123107149.43612959152.37333991156.18745633161.17371639167.77393SU



'u,a Jv,b Ju,c Jv,d

5 OVER 611 42,11 41,11 40,a 39,a 38,a 37,a 36,a 35,a 34,a 33,11 32,a 3i,a 30,

12 46,12 45,12 44,12 43,12 42,12 41,12 40,12 39,12 38,12 37,12 36,12 35,

13 49,13 48,13 47,13 46,13 45,13 44,13 43,13 42,13 41,13 40,

14 53,14 52,14 51,14 50,14 49,14 48,14 47,

(continuad)

16 4816 4716 4616 4516 4416 4316 4216 4116 4016 3916 3816 3716 36

17 5217 5117 5017 4917 4817 4717 4617 4517 4417 4317 4217 41

18 5518 54IS 5318 5213 5113 5018 4918 4813 4713 46

19 5919 5819 5719 5619 5519 5419 53

78.79.79.79.30.31.82.83.35.37.90.94.99.

9443936209049152369S088880422745422873592608322836304639787444956099370593139398391650826827913358432897

84.9822190085.0610077585.2523188485.5733521336.0449506586.6926103487.5478427288.6500423890.0490901791.8090513694.0135420796.77370544

91.0656217891.1873089691.4195061491.7781560692.2823795292.9553015593.8251476894.9267182596.3034012098.00996580

97.09582 76097.1609596897.3219240297.5912644597.9837333593.5170695699.21218053

1.630859733.766486066.025531428.42920229

11.0038091513.7823844716.8069518120.1317711623.3230836327.9912223932.7515802138.2920713544.37704092

0.575631752.643769914.S26371207.124432059.56236383

12.1639353514.9585715117.9831096621.2842713524.9221723023.9754208233.54872242

1.563170303.676130985.891692728.22463836

10.6929299713.3179830216.1262068719.1501463222.4304417526.01847554

0.501699932.562159274.711582276.961631359.32608435

11.8213294314.46700867

133.71423130133.88423466134.20884012134.71441356135.43353249136.40691417137.63610538139.33730769141.44593730144.12990614147.54230675151.90147929157.31397096

144.53173290144.72357363144.94642353145.32030860145.36934475146.62297779147.61749157148.39310715150.52189131152.56187602155.11300478158.30094647

155.60101554155.74291339156.C136494Ô156.43171479157.01925308157.80299357158.81543974150.09657339161.69608012163.67646731

166.50707578166.53309669156.77095430167.08524147167.54314312153.16503392168.97524333

152.56490051152.74021642153.07495321153.59629943154.33730523155.34141234156.66019300158.36227305150.53654309153.30114711156.81544138171.30556787177.08811655

153.48137125163.57532367153.30513786154.18934558164.75476174163.32015323166.53331075157.37C69453159.54091421171.6389417S174.26220125177.53957396

174.45133315174.59707299174.37512032175.30447471175.90736227176.71269545177.75233318179.06777326130.70995369182.74292419

135.35663222133.43457944135.62719573135.94943935136.41892243137.05652186137.83716662

Department of Mathematics

Pennsylvania State University

University Park, Pennsylvania 16802

1. T. C. Benton & H. D. Knoble, "Common zeros of two Bessel functions," Math. Comp., v. 32, 1978,

pp. 533-535.2. G. N. Watson, Treatise on Bessel Functions, Cambridge Univ. Press, Oxford, 1945.

3. Royal Society Mathematical Tables 7. Bessel Functions III (F. W. T. Olver, ed.), Cambridge

Univ. Press, Oxford, 1960.


Documents

Common Zeros of Two Bessel Functions Part II. Approximations … · 2018-11-16 · MATHEMATICS OF COMPUTATION VOLUME 41. NUMBER 163 JULY 1983, PAGES 203-217 Common Zeros of Two Bessel