Common Zeros of Two Bessel Functions Part II. Approximations … · 2018-11-16 · MATHEMATICS OF...
Transcript of Common Zeros of Two Bessel Functions Part II. Approximations … · 2018-11-16 · MATHEMATICS OF...
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 J^ix) 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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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 J^ix) 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 J^ix) 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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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 J^ix) 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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
COMMON ZEROS OF TWO BESSEL FUNCTIONS. PART II 209
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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:
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
COMMON ZEROS OF TWO BESSEL FUNCTIONS. PART II 213
«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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
COMMON ZEROS OF TWO BESSEL FUNCTIONS. PART II 217
'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.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use