Coefficients and Roots of the Polynomials which
Define the Derivatives of the Exponential
of (-e/T)
By Edwin S. Campbell, E. M. Fishbach,
and J. O. Hirschfelder
The w-th derivative
(1)
of exp (—t/T)
d" exp 1-e/rydT*
Wnlt/r\
can be shown to be of the form
= T~" exp l-e/T]WJ[t/T]
= ¿ bn,i[e/Ty.
The b„,i are constant coefficients in a homogeneous polynomial of the w-th order in
[e/7"]. These polynomials attain added importance from their relation to the Laguerre polynomials [1]
(2)
WJMT] = (-mL.it/T2
- «Ln-i[e/r]}
L„[e/r]:
the w-th order Laguerre polynomial.
The zeros of these polynomials are of interest since they locate the extreme values
of the derivatives of exp (—t/T). Furthermore, an accurate, simple computation
of the Wn for values of (t/T) which occur in physical problems sometimes requires
use of the polynomial roots. This occurs whenever the direct evaluation of Wn[t/T~\
by synthetic division introduces at intermediate steps of the calculation numbers
which are larger than the value of the polynomial. In such cases the subtraction of
these larger numbers can require the use of considerably more than the usual guard
figures which allow for the effect of ordinary round-off error. Fortunately the occurrence of this disastrous subtraction in synthetic division does not imply the occurrence when the polynomial is calculated using its roots
(3)
WJL</r\= fi &/Ï - r..,]
r„,j: the Z-th root of WnThe polynomial coefficients, bn,i, are integers. Table I lists all of their non-zero
digits which were computed from the recursion relations
(4)
bn,i = i-iy-H\
K.i = -in + l-
l)bn^i,i + Bb-t.t-i(2 < I < n - 1)
b»,n = 1.
The values in the Table were checked numerically
(5)_
in - l)bn,i = -»(«
by the distinct relations
- l)bn^,i.
Received 16 September 1957.
1
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
coefficients
Table I.
l
ßn.l
and roots
Exact Values of the Coefficients,bn_i
Pn.t
1+1
0
,
_,
¿2-4-1+1
n
n
"5
1+6
2-6
3+1
0
0
0
1 -2.4
2 +3-6
3
-I-2
4+l
1
1
1
0
1+12
2
_2'4
3 j-l'î
4_2'
5+1 T
?
%
7
7
0
U
1 -7.2
2 +1.8
3
-1.2
4+3
5-3
6+1
2
3
3
2
1
0
1
2
+5.04
-1.512
, , -,
3
4
4
I42
f
5
6
7
+6^3
-4.2
+1
1
2
3
4
5
6
7
8+1
-4.032
+1.4112
-1.4112
+5.88
-1.176
+1.176
-5.6
4
S
5
4
4
3
1
O«
9
1
2
3
4
5
+3.6288
-1.4515 2
+1.6934 4
-8.4672
+2.1168
S
6
6
S
5
7
ó
o
I2 0164
_7,°
.,
"•"
1
1
9
4
1
l
ßn,,
8
4
6
4
88
7
8
8
87
7
6
5
3
2
0
12
l
2
3
4
S
6
7
8
9
10
-4.7900
+2.6345
-4.3908
+3.2931
-1.3172
+3.0735
-4.3908
+3.9204
~2178
+7.26
16
088
48
36
544
936
48
8
9
9
9
9
8
7
6
5
3
H
12
-1-32
+1
13
14
1
2
3
4
5
6
7
8
+6.2270
-3.7362
+6.8497
-5.7081
+2.5686
-6.8497
+1.1416
-1.2231
9
10
11
12
13
+8.4942
-3.7752
1
2
3
4
5
6
7
208
1248
2288
024
4608
2288
2048
648
65
4
2
0
+1.0296
-1.56
+1
-8.7178
+5.6665
-1.1333
+1.0388
-5.1943
+1.5583
-2.9682
+3.7102
-3.0918
+1-7177
-6.2462
+1-4196
-1.82
+1
10
11
12
12
11
11
10
9
8
7
5
4
2
0
'
2
43
+1-3076 74368
-9.1537 20576
+1.9833
06124 8
-1.9833 06124
12
12
13
13
67
,
S
+10908
4,
7. ill2 88
+1.2700
77
89
-1.1130 79968
+1.08216108
TH!0.2 4
+6.3504
-6°48
+3-24
-?
+1
6
S
4
3
1
Ô
9
10
10
10
10
9
9
8
2912
88928
17785 6
74636 8
73184
11955 2
13248
6656
888
16
4
9
10
»
12
13
14
1S
2
0
T3-6288 £,
+í-6^
6
7
8
.?.
10
i>„,,
1 +3.9916
2
-1.9958
3 +2-9937
4
-1-9958
+6.9854
6 -1.3970
7 +1.6632
8-1.188
9 +4.95
10 -1.1
11 +1
2J.
^
< «8«
n
11
2
1
0
8
10
of polynomials
67
10
n
12
13
14
¡5
18368 64
-3.6360 59776
61228 8
+7.7915
-7.2144 072
+3.2792 76
-9.9372
+1.911
-2.1
+Ï"
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
12
11
11
10
8
7
S
5
4
0
coefficients
Table
I.
Exact
A.,1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
-2
+1
-3
+3
-2.
+8
-2
+3
-3
+2
-1
+5
-1
+2,
-2.
+1
0922 78988 8
5692 09241 6
6614 88230 4
9666 12249 6
3799 67349 76
7265 46949 12
0777 49273 6
3392 39904
7102 6656
8857 6288
5740 5248
9623 2
5288
52
4
and roots
of polynomials
3
Values of the Coefficients, &„,r—Continued
Pn.i
n
I
13
14
14
14
14
13
13
12
11
10
9
19
1
2
3
4
5
6
7
8
9
10
11
7
6
4
2
0
+3 .5568 74280 96
-2 .8454 99424 768
+7 .1137 48561 92
-8 .2993 73322 24
+5 3945 92659 456
-2 1578 37063 7824
+5 6514 78024 192
-1 0091 92504 32
+1 2614 90630 4
-1 1213 25004 8
+7 1357 04576
-3 2435 0208
+ 1 0395 84
-2 2848
+3 264
-2 72
+1
14
15
15
15
15
15
14
14
13
12
10
9
8
6
-6 .4023 73705 728
+5 .4420 17649 8688
-1 .4512 04706 63168
+ 1 .8140 05883 2896
-1 .2698 04118 30272
+5 5024 84512 64512
-1 5721 38432 18432
+3 0881 29063 2192
-4 2890 68143 36
+4 2890 68143 36
-3 1193 22286 08
+ 1 6541 86060
-6 3622 5408
+1 7478 72
-3 3292
+4 1616
-3 06
+1
15
16
17
17
12
13
14
15
16
17
18
19
20
6
7
8
9
10
11
12
13
14
15
16
4
2
0
17
16
16
15
14
13
12
11
9
8
6
4
2
0
1
2
3
4
5
17
18
19
20
21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
/3..i
+ 1.2164 51004 08832
-1.0948 05903 67948 8
+3.1019 50060 42521 6
-4.1359 33413 90028 8
+3.1019 50060 42521 6
-1.4475 76694 86510 08
+4.4805 94531 72531 2
-9.6012 73996 55424
+ 1.4668 61305 02912
-1.6298 45894 4768
+ 1.3335 10277 2992
-8.0818 80468 48
+3.6264 84825 6
-1.1955 44448
+2.8465 344
-4.7442 24
+5.2326
-3.42
+1
-2.4329 02008 17664
+2.3112 56907 76780 8
-6.9337 70723 30342 4
+9.8228 41858 01318 4
-7.8582 73486 41054 72
+3.9291 36743 20527 36
-1.3097 12247 73509 12
+3.0404 03432 24217 6
-5.0673 39053 73696
+6.1934 14399 01184
-5.6303 76726 3744
+3.8388 93222 528
-1.9686 63191 04
+7.5717 81504
-2.1633 66144
+4.5070 128
-6.6279 6
+6.498
-3.8
+1
+5.1090
-5.1090
+ 1.6178
-2.4268
+2.0627
-1.1001
+3.9291
-9.8228
+ 1.7735
-2.3647
+2.3647
-1.7914
+ 1.0335
-4.5430
+ 1.5143
-3.7858
+6.9593
-9.0972
+ 7.98
-4.2
+1
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
94217 17094 4
94217 17094 4
79835 43746 56
19753 15619 84
96790 18276 864
58288 09747 6608
36743 20527 36
41858 01318 4
68668 80793 6
58225 07724 8
58225 07724 8
83503 8464
48175 296
68902 4
56300 8
90752
58
17
18
18
18
18
18
17
16
16
15
14
12
11
10
8
6
4
2
0
18
19
19
19
19
19
19
18
17
16
15
14
13
11
10
8
6
4
2
0
19
20
21
21
21
21
20
19
19
18
17
16
15
13
12
10
8
6
4
2
0
coefficients
Table
I.
Exact
and roots of polynomials
Values of the Coefficients, b„,¡—Continued
I
ßn.l
Pn.l
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
-1.1240 00727 77760 768
+1.1802 00764 16648 8064
-3.9340 02547 22162 688
+6.2288 37366 43424 256
-5.6059 53629 79081 8304
+3.1767 07056 88146 37056
-1.210174116 90722 42688
+3.2415 37813 14435 072
-6.3029 90192 22512 64
+9.1043 19166 54740 48
-9.9319 84545 32444 16
+8.2766 53787 77036 8
-5.3055 47299 8528
+2.6236 22291 136
-9.9947 51585 28
+2.9151 35879 04
-6.4304 46792
+1.0507 266
-1.2289 2
+9.702
-4.62
+1
21
22
22
22
22
22
22
21
20
19
18
17
16
15
13
12
10
9
7
4
2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
+2.5852
-2.8437
+9.9530
-1.6588
+1.5758
-9.4553
+3.8271
-1.0934
+2.2780
-3.5436
+4.1879
-3.8072
+2.6846
-1.4750
+6.3216
-2.1072
+5.4230
-1.0633
+1.5545
-1.6364
+1.1688
-5.06
+1
01673 88849 7664
21841 27734 74304
26444 47071 60064
37740 74511 93344
95853 70786 33676 8
75122 24718 02060 8
75644 71909 67500 8
78755 63402 76428 8
80740 90422 4256
81152 51768 2176
86816 61180 6208
60742 37436 928
06933 72551 68
58754 79424
80377 6896
26792 5632
10127 92
35319 2
838
04
6
22
23
23
24
24
23
23
23
22
21
20
19
18
17
15
14
12
11
9
7
5
2
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
-6.2044
+7.1351
-2.6162
+4.5783
-4.5783
+2.8996
-1.2427
+3.7725
-8.3833
+1.3972
-1.7782
+1.7513
-1.3471
+8.1423
-3.8772
+1.4539
-4.2764
+9.7826
-1.7162
+2.2582
84017 33239 43936
56619 93225 35526 4
24093 97515 96359 68
92164 45652 93629 44
92164 45652 93629 44
48370 82246 85965 312
06444 63820 08270 848
01706 93739 53679 36
37126 52754 52620 8
22854 42125 75436 8
83632 89978 23283 2
39941 49220 98688
84570 37862 2976
24326 46420 48
97298 31628 8
86486 86860 8
30843 7312
84936 64
60515 2
3752
23
24
25
25
25
25
25
24
23
23
22
21
20
18
17
16
14
12
11
9
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
coefficients
Table
«
I.
Exact
and roots
of polynomials
5
Values of the Coefficients, bn,t—Continued
l
ßn.l
Pn.l
24
21
22
23
24
-2.1507 024
+1.3965 6
-5.52
+1
25
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
+1.5511 21004 33309 85984
-1.8613 45205 19971 83180 8
+7.1351 56619 93225 35526 4
-1.308112046 98757 98179 84
+1.3735 17649 33695 88088 832
-9.1567 84328 91305 87258 88
+4.1423 54815 46066 94236 16
-1.3314 71190 68378 66004 48
+3.1437 51422 44782 94732 8
-5.5888 91417 68503 01747 2
+7.6212 15569 57049 56928
-8.0831 07422 27173 7856
+6.7359 22851 89311 488
-4.4412 67814 43502 08
+2.3263 78378 98977 28
-9.6932 43245 79072
+3.2073 23132 7984
-8.3851 58517 12
+1.7162 60515 2
-2.7098 85024
+3.2260 536
-2.7931 2
+1.656
-6
+1
25
26
26
27
27
26
26
26
25
24
23
22
21
20
19
17
16
14
13
11
9
7
5
2
0
26
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-4.0329
+5.0411
-2.0164
+3.8648
-4.2513
+2.9759
-1.4171
+4.8080
-1.2020
+2.2704
-3.3025
+3.7528
-3.3679
+2.4056
-1.3746
+6.3006
-2.3164
+6.8129
-1.5936
+2.9357
-4.1938
+4.5388
-3.588
+1.95
-6.5
+1
14611 26605 63558
43264 08257 04448
57305 63302 81779
76502 46330 40076
64152 70963 44084
54906 89674 40859
21384 23654 48028
90410 80256 27238
22602 70064 06809
87138 43454 35084
26746 81388 14668
71303 19759 2576
61425 94655 744
86732 81896 96
78133 03941 12
08109 76396 8
00040 3544
41295 16
70478 4
08776
6968
2
26
27
28
28
28
28
28
27
27
26
25
24
23
22
21
19
18
16
15
13
11
9
7
5
2
0
1
2
3
4
5
6
7
8
9
+1.0888
-1.4155
+5.8981
-1.1796
+1.3565
-9.9481
+4.9740
-1.7764
+4.6878
86945 04183
53028 55438
37618 97660
27523 79532
71652 36461
92117 34054
96058 67027
62878 09652
88150 53249
27
7
5
2
0
4
2
8
48
136
16
4
6
8
8
52160 768
57808 9984
74204 16
14840 832
97066 9568
45157 6832
22578 8416
58063 872
86557 44
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
28
29
29
30
30
29
29
29
28
coefficients
Table
I.
Exact
and roots
of polynomials
Values of the Coefficients, 6„,¡—Continued
I
ßn.l
Pn.l
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
-9.3757
+1.4489
-1.7563
+1.6887
-1.2990
+8.0418
-4.0209
+1.6261
-5.3140
+1.3984
-2.9440
+4.9068
-6.3725
+6.2969
-4.563
+2.2815
-7.02
+1
76301 06499 73114 88
83610 16459 04935 936
43769 89647 33255 68
92086 43891 66592
70835 72224 3584
67078 28055 552
33539 14027 776
12828 32878 88
94210 2248
45844 796
96515 36
27525 6
0328
4
27
27
26
25
24
22
21
20
18
17
15
13
11
9
7
5
2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
-3.0488
+4.1159
-1.7835
+3.7158
-4.4589
+3.4185
-1.7906
+6.7150
-1.8652
+3.9378
-6.4437
+8.2987
-8.5115
+7.0149
-4.6766
+2.5331
-1.1175
+4.0174
-1.1746
+2.7821
-5.2993
+8.0293
-9.5209
+8.6240
-5.7493
+2.6535
-7.56
+1
83446 11713 86050 1504
92652 25813 71167 70304
96815 97852 60839 33798 4
26699 95526 26748 6208
92039 94631 52098 34496
60563 95884 16608 73113 6
74581 12129 80128 38297 6
29679 20486 75481 43616
86022 00135 20967 0656
26046 44729 88708 2496
15348 73194 36068 0448
24312 76083 64633 088
12115 65213 99623 68
82512 90011 53536
55008 60007 69024
88129 65837 49888
82998 37869 4848
55222 92994 88
94509 62864
71207 0152
73727 648
54132 8
7328
7
8
6
29
30
31
31
31
31
31
30
30
29
28
27
26
25
24
23
22
20
19
17
15
13
11
9
7
5
2
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
+8.8417
-1.2378
+5.5703
-1.2069
+1.5086
-1.2069
+6.6092
-2.5964
+7.5730
-1.6829
+2.9068
-3.9638
+4.3195
-3.7974
+2.7124
-1.5822
+7.5623
-2.9656
+9.5385
61993 73970 19545 43616
46679 12355 82736 36106 24
10056 05601 22313 62478 08
00512 14546 93167 95203 584
25640 18183 66459 94004 48
00512 14546 93167 95203 584
17090 32042 72110 21352 96
78142 62588 21186 15531 52
61249 32548 95126 28633 6
02499 85010 87805 84140 8
31590 65018 78937 36243 2
61259 97752 89460 03968
92398 69346 10309 0176
43866 98326 24447 488
59904 98804 46033 92
68277 90969 26853 12
11622 36250 18048
12400 92647 1296
19418 18455 68
30
32
32
33
33
33
32
32
31
31
30
29
28
27
26
25
23
22
20
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
coefficients
Table
»
I.
Exact
7
and roots of polynomials
Values of the Coefficients, bn,i—Continued
/
ßn.i
P..I
29
20
21
22
23
24
25
26
27
28
29
-2.5101
+5.3788
-9.3140
+1.2885
-1.4005
+1.1671
-7.1823
+3.0693
-8.12
+1
36688 99593 6
64333 56272
50794 048
05050 56
48968
2414
024
6
19
17
15
14
12
10
7
5
2
0
30
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
-2.6525
+3.8461
-1.7948
+4.0384
-5.2500
+4.3750
-2.5000
+1.0267
-3.1374
+7.3206
-1.3310
+1.9158
-2.2106
+2.0648
-1.5732
+9.8326
-5.0609
+2.1500
-7.5441
+2.1838
-5.1995
+1.0129
-1.6014
+2.0307
-2.0307
+1.5621
-8.9011
+3.5322
-8.7
+1
28598 12191 05863 63084 8
66467 27677 03502 26472 96
77684 72915 94967 72354 048
74790 64060 88677 37796 608
17227 83279 15280 59135 5904
14356 52732 62733 82612 992
08203 72990 07276 47207 424
89083 67478 06559 97960 192
11089 00627 42266 60433 92
25874 34797 31955 41012 48
22886 24508 60355 52911 36
66275 65580 56572 35251 2
14933 44900 65275 79136
60102 67214 89543 3216
26744 89306 58699 6736
67155 58166 16872 96
31624 19644 35155 2
68990 67169 16896
01721 65505 856
18919 42646 432
68855 77729 6
03023 85272
27705 696
96003 6
96003 6
50772
44
32
33
34
34
34
34
34
34
33
32
32
31
30
29
28
26
25
24
22
21
19
18
16
14
12
10
7
5
2
0
Legend: Wn(x) = 2 b„,, xl
i-i
b..t = ßn,I X ÎO*"".',
1 g ft.,1 < 10
Table
n
I
II.
Roots of Wn (t/T)
rn_i
C
e„,i
2
1
2.00000 00000 0000
0
0
3
3
1
2
1.26794 91924 3112
4.73205 08075 6888
0
0
5
5
4
4
4
1
2
3
0.93582 22275 2409
3.30540 72893 3227
7.75877 04831 4363
0
0
0
5
5
5
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
coefficients
8
Table
»
5
5
5
5
/
8
8
8
8
8
8
8
1
2
3
4
1
2
3
4
5
1
2
3
4
5
6
1
2
3
4
5
6
7
9
9
9
9
9
9
9
9
II.
and roots
of polynomials
Roots of Wn (t/T)—Continued
r„,i
0.46102
1.56358
3.35205
5.91629
9.42069
14.19416
21.09217
19279 8143
50076 4627
87516 8905
43126 8321
08532 7825
59585 7838
31510 1760
69712 0486
30659 2081
81217 1117
98096 4345
15204 7699
65667 0471
04290 8311
35523 8068
42198 0496
61896 5431
05025 3674
72490 2042
93830 2156
55480 0748
69079 5447
1
2
3
4
5
6
7
8
0.40938
1.38496
2.95625
5.18194
8.16170
12.07005
17.24973
24.58595
10
10
10
10
10
10
10
10
10
1
2
3
4
5
6
7
8
9
11
11
11
11
11
11
11
11
11
11
1
2
3
4
5
6
7
8
9
10
6
6
6
6
6
0.74329
2.57163
5.73117
10.95389
C
e„,i
0
0
0
1
5
5
50
6
0
0
0
0
1
0
0
0
0
1
1
50
50
50
50
10
50
50
50
50
6
16
0
0
0
0
0
1
1
50
50
50
50
50
10
22
35732 0319
31848 0312
45561 6887
31010 4007
96881 4582
51268 3715
55261 4898
52436 5281
0
0
0
0
0
1
1
1
5
5
5
5
5
7
15
30
0.36817
1.24335
2.64603
4.61688
7.22178
10.56732
14.83591
20.38218
28.11834
84529 4174
79621 4047
38413 8420
25146 3485
65393 9663
08077 4184
45152 6107
19854 4899
33810 4993
0
0
0
0
0
1
1
1
1
5
5
5
50
50
56
110
207
39
0.33452
1.12825
2.39586
4.16684
6.48735
9.42835
13.10172
17.69648
23.57778
31.68280
86763 2476
33558 7666
99247 4731
09879 2878
30313 8081
48133 3561
35803 6780
75668 4621
70883 6019
09748 3192
0
0
0
0
0
0
1
1
1
1
5
5
5
5
5
5
9
16
28
50
0.61703
2.11296
4.61083
8.39906
14.26010
0.52766
1.79629
3.87664
6.91881
11.23461
17.64596
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
coefficients
Table
'
II.
and roots
9
of polynomials
Roots of Wn (t/T)—Continued
r».i
C
12
12
12
12
12
12
12
12
12
12
12
1
2
3
4
5
6
7
8
9
10
11
0.30652
1.03279
2.18961
3.79904
5.89491
8.52729
11.77100
15.74226
20.63580
26.82634
35.27439
67021 3005
73987 7972
19419 6840
76060 5497
11715 0275
20080 7027
65654 9068
02870 2615
56686 4129
99493 7082
07009 6510
0
0
0
0
0
0
5
5
5
5
5
5
7
12
213
36
62
13
13
13
13
13
13
13
13
13
13
13
13
1
2
3
4
5
6
7
8
9
10
11
12
0.28285
0.95232
2.01649
3.49235
5.40549
7.79281
10.70738
14.22715
18.47199
23.64178
30.12005
38.88928
83482
60413
21385
40697
10200
39404
86889
23637
66342
37524
86261
43760
3992
6462
7771
7799
1572
1242
8909
8996
2982
0122
0650
9550
0
0
0
0
0
0
5
5
50
50
5
5
6
10
171
279
455
76
14
14
14
14
14
14
14
14
14
14
14
14
14
1
2
3
4
5
6
7
8
9
10
11
12
13
0.26258
0.88355
1.86903
3.23241
4.99357
7.18061
9.83280
13.00562
16.77961
21.27791
26.70503
33.45278
42.52444
83981 7108
03073 8774
38151 9979
86994 5120
56070 7419
04941 0079
82510 2000
24002 5865
36411 7585
17554 4831
40989 2773
49657 6721
75660 1840
0
0
0
0
0
0
0
5
5
50
15
15
1
2
3
4
5
6
0.24503
0.82408
1.74187
3.00913
4.64163
6.66134
9.09830
11.99370
15.40498
19.41499
24.14975
29.81810
36.81962
46.17743
30150 9310
22200 4784
24007 6449
24590 1792
40665 9064
14966 2717
27037 7053
34331 8574
80310 5145
28281 6615
79520 2928
51196 7132
42569 9721
00169 8345
0
0
0
0
0
0
0
15
15
15
15
15
15
15
15
15
15
15
15
7
8
9
10
11
12
13
14
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
50
5
5
5
85
141
226
357
559
91
5
50
50
50
50
5
5
7
119
1890
2920
4460
6760
1060
coefficients
Table
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
of polynomials
Roots of Wn (t/T)~ -Continued
C
rn.l
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.22968
0.77214
1.63105
2.81514
4.33716
6.21464
8.47116
11.13833
14.25891
17.89205
22.12262
27.07931
32.97497
40.21658
49.84622
05054 2513
49103 7539
30990 6745
45900 1225
40773 3756
27645 5924
39813 4671
19657 5080
00216 2446
34381 6953
01748 2668
14990 4742
35524 0162
37114 8568
17085 5745
0
0
0
0
0
0
0
1
2
3
4
5
6
03052 3945
82432 5183
31603 7353
09986 1195
81608 8018
55151 0563
41853 0666
82899 5104
10707 0697
32168 6805
60200 3371
70253 3575
29201 6784
45436 3216
51841 5001
11602 5465
0
0
0
0
0
0
0
12
13
14
15
16
0.21614
0.72638
1.53359
2.64497
4.07097
5.82585
7.92850
10.40380
13.28466
16.61517
20.45600
24.89384
30.05986
36.17069
43.64036
53.52915
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
0.20410
0.68576
1.44719
2.49443
3.83615
5.48411
7.45372
9.76498
12.44386
15.52434
19.05171
23.08800
27.72155
33.08585
39.40115
47.08820
57.22481
91085 7933
75894 9453
86793 8049
08859 6248
60319 9932
54208 4047
29495 8128
43667 5180
10519 8395
30012 1457
87593 8568
07008 7174
63462 7824
98280 6407
37933 0861
96546 0141
18319 1492
0
0
0
0
0
0
0
0
7
8
9
10
11
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
II.
and roots
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
e.,i
5
5
50
50
5
5
5
6
102
1600
2450
3680
5450
8060
1240
5
50
50
50
50
50
5
5
88
1380
2090
30900
45000
65600
9540
14300
5
5
50
50
50
5
5
5
77
1200
1810
26700
38500
54800
77500
1 11000
16300
coefficients
Table
I
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
21
21
21
21
21
21
1
2
3
4
5
6
21
21
21
21
21
21
21
21
8
9
10
11
12
13
14
17
18
19
7
II.
and roots
11
of polynomials
Roots of Wn (t/T)—Continued
en,i
?n,l
0.19334
0.64946
1.37007
2.36027
3.62737
5.18117
7.03443
9.20350
11.70932
14.57876
17.84675
21.55965
25.78070
30.59981
36.15265
42.66288
50.55778
60.93200
77686 7901
18107 9655
54867 1706
76257 7508
58656 5560
49562 2015
06230 6566
66842 8214
47926 7879
11306 9283
98274 1882
87534 0514
10050 5240
44865 1918
02790 4671
77797 8564
33977 1778
77265 0073
0
0
0
0
0
0
0
0
1
1
1
2
2
2
2
2
2
2
5
5
5
5
5
5
5
5
69
1060
15900
12
17
23
33
45
64
93
0.18366
0.61681
1.30079
2.23994
3.44047
4.91064
6.66116
8.70559
11.06110
13.74939
16.79812
20.24308
24.13156
28.52794
33.52361
39.25629
45.95295
54.04709
64.64971
51730 9616
63821 2673
94029 6677
93990 0886
19428 8150
58796 7473
06120 5821
71771 2802
86835 7575
41200 8341
30059 5300
49633 0246
72676 0267
79713 1414
76172 8890
42907 8263
06486 3834
30247 0194
24378 1605
0
0
0
50
50
500
500
500
50
50
50
612
9450
14100
10
15
20
28
39
53
0.17490
0.58730
1.23822
2.13139
3.27213
4.66749
6.32653
8.26067
10.48416
13.01484
15.87508
19.09325
22.70589
26.76117
67523 8661
30806 3812
51018 3420
62600 7712
31335 1677
44658 8837
61976 7362
09520 1371
73812 0829
87721 5828
70127 1499
19076 0624
38881 7321
02293 7952
0
0
0
0
0
1
1
1
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
73
105
50
50
500
500
500
500
500
50
55
8470
1 26000
18
26
357
coefficients
Table
II.
and roots
of polynomials
Roots of Wn (t/T)- -Continued
C
fn.l
21
21
21
21
21
21
15
16
17
18
19
20
31.32451
36.48870
42.39342
49.26881
57.55442
68.37703
61370 0729
33461 4824
27457 7640
38498 6849
09713 1555
78145 5231
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
1
2
3
4
5
6
61390 1971
08612 5406
07583 5136
74520 5904
20402 1490
50426 1758
21681 1509
77586 0933
88709 4385
86988 9685
02175 9276
77154 4042
84662 6343
63492 4893
42999 0910
02974 6011
82536 2833
99787 8368
92435 3047
57571 7212
96312 7984
0
0
0
0
0
0
0
0
0
19
20
21
0.16694
0.56049
1.18142
2.03295
3.11969
4.44769
6.02471
7.86043
9.96688
12.35891
15.05493
18.07788
21.45678
25.22887
29.44311
34.16593
39.49140
45.56112
52.60828
61.07827
72.11320
23
23
23
23
23
23
23
23
23
23
23
23
23
23
23
23
23
23
23
23
23
23
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
0.15967
0.53602
1.12962
1.94327
2.98097
4.24798
5.75098
7.49829
9.50013
11.76904
14.32037
17.17304
20.35054
23.88244
27.80661
32.17279
37.04832
42.52855
48.75689
55.96946
64.61735
75.85754
90124 6359
44507 2533
03228 4296
26194 9229
42342 9980
60975 3645
94501 8811
71042 4725
87408 1857
61305 3152
80923 6802
70973 6204
65523 0563
31429 7127
96348 6417
15883 7417
56128 1149
84501 9839
07697 4619
93367 7959
31062 9634
84527 5272
0
0
0
0
0
0
0
0
0
7
8
9
10
11
12
13
14
15
16
17
18
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
490
666
898
1210
1650
235
50
50
500
500
500
500
50
50
50
765
11300
63000
230
318
4310
5830
7810
10400
13900
1860
259
50
50
500
500
500
500
500
50
5
69
1030
14700
10
284
3850
5170
68600
90600
19000
15600
20800
2870
coefficients
Table
I
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
II.
and roots
of polynomials
13
Roots of Wn (t/T)— Continued
r..i
C
0.15301
0.51360
1.08218
1.86121
2.85418
4.06569
5.50154
7.16891
9.07661
11.23531
13.65799
16.36046
19.36209
22.68683
26.36479
30.43444
34.94620
39.96833
45.59738
51.97849
59.35068
68.17050
79.60945
84895 9942
83707 0681
80942 3753
64108 9897
83431 3952
47966 6047
23910 2483
89382 1722
44660 1974
54871 5978
92756 6470
86713 7067
01881 5387
57294 7187
43150 0290
26913 9403
46462 8518
92266 7395
43134 3094
53322 1339
02360 3490
51810 0240
44050 8424
0
0
0
0
0
0
0
0
0
16257 8146
48461 2182
20353 8078
95467 3969
45322 0507
82901 4203
15403 0644
56382 7667
61931 4098
14241 0897
36754 8216
13581 6414
24144 4275
28735 8341
20017 9402
24031 5677
73051 8389
10435 6086
51864 7370
32211 6803
77787 5012
41754 7154
59637 8361
49267 2194
0
0
0
0
0
0
0
0
0
67262 5809
45378 8442
34056 2151
16871 9240
0
0
0
0
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
17
18
19
20
21
22
23
24
0.14689
0.49299
1.03859
1.78584
2.73784
3.89860
5.27322
6.86794
8.69039
10.74977
13.05715
15.62591
18.47224
21.61592
25.08136
28.89913
33.10826
37.75986
42.92302
48.69545
55.22399
62.75044
71.73671
83.36839
26
26
26
26
1
2
3
4
0.14123
0.47397
0.99838
1.71638
1
1
1
1
5
5
50
50
50
50
50
5
5
63
933
13400
88000
2580
3470
46300
61100
80000
04000
36000
77000
23100
3140
5
5
50
50
50
50
50
50
5
58
852
12200
1 71000
2340
3140
21
27
36
46
59
76
98
129
174
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
5
5
50
50
coefficients
Table
»
I
II.
and roots of polynomials
Roots of Wn (t/T)—Continued
r„,i
C
e»,i
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2.63069
3.74487
5.06340
6.59177
8.33662
10.30594
12.50927
14.95806
17.66600
20.64967
23.92920
27.52942
31.48133
35.82451
40.61069
45.90978
51.82061
58.49167
66.16744
75.31508
87.13389
31145 8470
77262 0273
83123 3855
56068 7320
63598 0514
30256 1368
80113 1609
12826 7794
89930 4484
47456 6110
78044 9273
09021 3584
78942 1101
67628 4751
00156 5943
68582 2976
58754 0514
48142 7643
93598 1048
13581 0590
48199 8148
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
50
50
50
50
5
5
78
11200
1 56000
2140
14
19
25
32
414
526
672
854
110
141
189
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
0.13600
0.45636
0.96118
1.65214
2.53167
3.60294
4.86990
6.33740
8.01127
9.89850
12.00738
14.34778
16.93144
19.77239
22.88750
26.29727
30.02688
34.10778
38.58003
43.49597
48.92632
54.97096
61.77999
69.60052
78.90479
90.90552
12570 4444
93581 3000
10273 0921
28929 8375
76794 9454
55780 8908
60826 2376
31557 2642
62543 8048
34830 7608
61381 8993
78810 5506
76204 2762
41164 6035
43962 7852
10029 7123
18438 7846
47142 1721
51906 4629
58235 1797
26588 5339
30253 0608
23077 3237
54573 4751
29547 6498
80995 1289
0
0
0
0
0
0
0
0
0
0
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
5
5
50
50
50
50
50
50
5
5
72
10300
143
1950
13
17
23
292
374
472
598
758
952
1210
156
206
28
28
28
28
1
2
3
4
0.13114
0.44002
0.92665
1.59256
02048 4340
68410 2122
89979 7975
14353 9651
0
0
0
0
5
5
5
5
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
coefficients
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
28
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
and roots
Table
I
II.
5
6
7
8
9
2.43989 88598 8214
3.47148 98767 8727
4.69085 73686 2107
6.10229 80663 5926
7.71097 27209 1486
9.52302 14214 8264
11.54571 07983 2683
13.78762 26083 0963
16.25889 71483 2334
18.97155 08191 9327
21.93989 61709 5973
25.18110 69743 2692
28.71599 40224 8103
32.57009 65076 4767
36.77526 28337 6414
41.37202 26709 0563
46.41330 39270 2954
51.97058 20538 3640
58.14479 02456 7904
65.08757 79811 6239
73.04862 35108 1621
82.50512 46756 5939
94.68291 12582 9295
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
15
of polynomials
Roots of Wn (t/T)—Continued
r».i
0.12661 47772 6314
0.42481 56894 2803
0.89453 69967 2982
1.53714 63597 6714
2.35458 60074 0362
3.34938 08943 4731
4.52468 00068 0930
5.88431 36574 5907
7.43286 73847 3363
9.17577 57820 5631
11.11944 12276 9518
13.27138 44455 9570
15.64043 65527 0260
18.23698 62111 4301
21.07330 14271 4752
24.16395 46279 8314
27.52639 39971 5279
31.18172 74331 4674
35.15582 50255 9695
39.48091 56524 1023
44.19798 25434 6970
49.36051 71053 0558
55.04072 95577 5849
61.34056 90221 6674
68.41319 89167 3268
76.51079 93357 6897
86.11542 16992 4401
98.46569 76629 3931
e».i
0
0
0
0
0
0
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
5
5
5
5
5
5
7
10
132
1800
12
16
21
266
338
430
5380
6760
8480
10600
1330
171
224
5
5
5
5
5
5
5
5
5
5
6
9
123
1670
11
15
19
244
308
3900
4880
6100
7580
9440
11700
14700
1860
242
16
coefficients
Table
I
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
II.
and roots of polynomials
Roots of Wn (t/T)- -Continued
0.12239
0.41062
0.86457
1.48547
2.27507
3.23564
4.37001
5.68161
7.17453
8.85361
10.72452
12.79390
15.06949
17.56035
20.27709
23.23221
26.44052
29.91973
33.69122
37.78117
42.22218
47.05567
52.33567
58.13511
64.55692
71.75574
79.98619
89.73509
102.25357
25
26
27
28
29
C
r*,i
13636 8232
22202 4781
25660 9784
36509 1174
48206 5174
71256 6498
73777 7174
56855 1702
65183 2250
57459 8089
73673 4938
50612 0659
56117 3280
39915 4328
38755 2512
33320 6006
46096 4200
14194 9155
07222 7026
59363 7412
88741 5065
82251 9296
93797 4686
50396 8783
28253 1261
28866 9527
96989 1509
12115 8820
28564 7799
5
5
5
5
5
5
5
5
5
5
6
8
11
1540
10
14
18
224
284
3580
4440
55200
68400
84800
1 04000
12900
16000
2020
260
0
0
0
0
0
0
0
0
0
0
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Legend:
r»,i: the Ith root of the n-th order polynomial,
W„. The estimated
error bound is
| r„., - root | g e„,, X 10-»
The significance of
0: the root was
1: the root was
2: the root was
the numbers in column C is:
computed explicitly
computed as rn,i = 10/V„,i
computed as rn.i = 20/f„j
The formulae of equation (4) can be verified by
formal construction of successive derivatives of exp
tions have been established they can be used with a
establish the correctness of explicit formulae which
poses, albeit not for calculations
,,,
(6)
r
(-!)-'/(/+
*-•=-prnn-•
l)t
...
induction based on the actual
(—t/T) [1]. After these relasimple induction argument to
are convenient for other pur-
(n-
l)'n
The roots were located by the method of false position [2], which depends upon
evaluation of the polynomials for successive approximations to the roots. The evaluations were computed by synthetic division using double precision arithmetic on
the ORDVACcomputer at the Aberdeen Proving Ground. The polynomials were
calculated in one of three forms as described in the legend to Table II. This was
necessary since the original form of the polynomial was relatively insensitive to
changes in approximations for certain roots; that is, the remainder was too small
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17
COEFFICIENTS AND ROOTS OF POLYNOMIALS
Table
III.
Symmetric Function Check on Roots
n
in.n-l
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
2
6
12
20
30
42
56
72
90
110
132
156
182
210
240
272
306
342
380
420
462
506
552
600
650
702
756
812
870
dn
0
0
1
4
10
+11
6
+1
28
+
5
+
20
+
47
+
94
- 371
- 1263
-13059
+21289
+ 927
+
27
- 6267
+39096
- 2411
-55267
- 5915
+23010
+ 1762
+ 223
- 4329
-18880
Legend: fc„.„_i - 2 r„,¡ = dn X 10"»
i=i
compared with the largest term in the synthetic division schema. When this occurred, one of the other forms of the polynomial was used in order to define the root
more accurately.
The error estimates of Table II were based upon locating the root between two
approximate values which gave remainders of opposite sign. Although this method
of locating roots is inherently self-checking, a further numerical test was made by
computing the symmetric function summations of Table III.
Edwin S. Campbell
New York University,
New York
E. M. Fischbach
Ballistic Research
Laboratories,
Aberdeen, Maryland
J. O. Hirschfelder
University of Wisconsin
Milwaukee, Wisconsin
1. J. O. Hirschfelder
Propagation
(CM-784),
17 ff.
2. F. B. Hildebrand,
& Edwin S. Campbell, Analytical (power series) Solutions of Flame
University of Wisconsin Naval Research Laboratory,
May 27, 1953, p.
Introduction
to Numerical Analysis,
McGraw-Hill
1956, p. 446.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Book Co., New York,