Advances in Mathematical Physics

Research Article

An Alpha-Beta Phase Diagram Representation of the Zeros and Properties of the Mittag-Leffler Function

Table 3

Reference values in phase .
(a) Reference value transition points along the line
TransitionAlpha
1 to 3 zeros1.3395747103
3 to 5 1.4892313259
5 to 7 1.5725808412
7 to 9 1.6280321346
9 to 11 1.6684183137
11 to 13 1.6995219670
13 to 151.7244096473
15 to 17 1.7448875732
17 to 19 1.7621012104
19 to 21 1.7768180976
21 to 23 1.7895749696
23 to 25 1.8007601937
25 to 271.8106627476
27 to 29 1.8195027301
29 to 311.8274511290
31 to 33 1.8346430576
33 to 35 1.8411868715
35 to 37 zeros1.8471706026
37 to 39 1.8526665973
39 to 41 1.8577349238
41 to 431.8624259162
43 to 451.8667821039
45 to 471.8708396938
47 to 491.8746297224
49 to 511.8781789624
51 to 531.8815106427
53 to 551.8846450251
55 to 571.8875998709
57 to 591.8903908216
59 to 611.8930317103
61 to 631.8955348200
63 to 651.8979110982
65 to 671.9001703362
67 to 691.9023213199
69 to 71 zeros1.9043719579
71 to 731.9063293896
73 to 751.9082000780
75 to 771.9099898892
77 to 791.9117041604
79 to 811.9133477593
81 to 831.9149251354
83 to 851.9164403642
85 to 871.9178971868
87 to 891.9192990435
89 to 911.9206491039
91 to 931.9219502937
93 to 951.9232053177
95 to 971.9244166806
97 to 991.9255867056
99 to 1011.9267175505
101 to 1031.9278112225
(b) Reference value transition points along the horizontal line
TransitionAlpha
1 to 3 zeros1.4221906908
3 to 5 1.5718839229
5 to 7 1.6490682373
7 to 9 1.6985162237
9 to 11 1.7336930327
11 to 13 1.7603388117
13 to 151.7813926516
15 to 17 1.7985433447
17 to 19 1.8128419490
19 to 21 1.8249822706
21 to 23 1.8354435176
23 to 25 1.8445688178
25 to 271.8526111866
27 to 29 1.8597618108
29 to 311.8661681768
31 to 33 1.8719460965
33 to 35 1.8771879211
35 to 37 zeros1.8819682945
37 to 39 1.8863482727
39 to 41 1.8903783311
41 to 431.8941005978
43 to 451.8975505379
45 to 471.9007582408
47 to 491.9037494173
49 to 511.9065461804
51 to 531.9091676623
53 to 551.9116305080
55 to 571.9139492725
57 to 591.9161367439
59 to 611.9182042070
61 to 631.9201616614
63 to 651.9220180019
65 to 671.9237811690
67 to 691.9254582752
69 to 71 zeros1.9270557120
71 to 731.9285792400
73 to 751.9300340659
75 to 771.9314249085
77 to 791.9327560555
79 to 811.9340314117
81 to 831.9352545420
83 to 851.9364287076
85 to 871.9375568985
87 to 891.9386418610
89 to 911.9396861230
91 to 931.9406920148
93 to 951.9416616889
95 to 971.9425971365
97 to 991.9435002028
99 to 1011.9443725998
101 to 1031.9452159190
(c) Reference value transition points along the line
TransitionAlpha
2 to 4 zeros1.2916629732
4 to 6 1.4337910571
6 to 8 1.5175792520
8 to 10 1.5751778645
10 to 12 1.6180496252
12 to 14 1.6515923598
14 to 161.6787594800
16 to 18 1.7013316816
18 to 20 1.7204590356
20 to 22 1.7369236242
22 to 24 1.7512791072
24 to 26 1.7639303013
26 to 281.7751811559
28 to 30 1.7852650145
30 to 321.7943644328
32 to 34 1.8026245763
34 to 361.8101625260
36 to 38 zeros1.8170738953
38 to 40 1.8234376334
40 to 42 1.8293195759
42 to 441.8347751111
44 to 461.8398512130
46 to 481.8445880105
48 to 501.8490200131
50 to 521.8531770789
52 to 541.8570851871
54 to 561.8607670588
56 to 581.8642426610
58 to 601.8675296189
60 to 621.8706435553
62 to 641.8735983713
64 to 661.8764064818
66 to 681.8790790112
68 to 701.8816259606
70 to 72 zeros1.8840563472
72 to 741.8863783254
77 to 761.8885992882
76 to 781.8907259561
78 to 801.8927644526
80 to 821.8947203703
82 to 841.8965988280
84 to 861.8984045205
86 to 881.9001417625
88 to 901.9018145263
90 to 921.9034264764
92 to 941.9049809984
94 to 961.9064812261
96 to 981.9079300641
98 to 1001.9093302094
100 to 1021.9106841692
102 to 1041.9119942782
(d) Reference value transition points along the horizontal line
TransitionAlpha
2 to 4 zeros1.3395747103
4 to 6 1.4892313259
6 to 8 1.5725808412
8 to 10 1.6280321346
10 to 12 1.6684183137
12 to 14 1.6995219670
14 to 161.7244096473
16 to 18 1.7448875732
18 to 20 1.7621012104
20 to 22 1.7768180976
22 to 24 1.7895749696
24 to 26 1.8007601937
26 to 281.8106627476
28 to 30 1.8195027301
30 to 321.8274511290
32 to 34 1.8346430576
34 to 361.8411868715
36 to 38 zeros1.8471706026
38 to 40 1.8526665973
40 to 42 1.8577349238
42 to 441.8624259162
44 to 461.8667821039
46 to 481.8708396938
48 to 501.8746297224
50 to 521.8781789624
52 to 541.8815106427
54 to 561.8846450251
56 to 581.8875998709
58 to 601.8903908216
60 to 621.8930317103
62 to 641.8955348200
64 to 661.8979110982
66 to 681.9001703362
68 to 701.9023213199
70 to 72 zeros1.9043719579
72 to 741.9063293896
77 to 761.9082000780
76 to 781.9099898892
78 to 801.9117041604
80 to 821.9133477593
82 to 841.9149251354
84 to 861.9164403642
86 to 881.9178971868
88 to 901.9192990435
90 to 921.9206491039
92 to 941.9219502937
94 to 961.9232053177
96 to 981.9244166806
98 to 1001.9255867056
100 to 1021.9267175505
102 to 1041.9278112225
(e) Reference value transition points along the horizontal line
TransitionAlpha
2 to 4 zeros1.7100323859
4 to 6 1.7651251779
6 to 8 1.7997959826
8 to 10 1.8241639279
10 to 12 1.8424530845
12 to 14 1.8567978278
14 to 161.8684122443
16 to 18 1.8780456420
18 to 20 1.8861890746
20 to 22 1.8931795541
22 to 24 1.8992570236
24 to 26 1.9045975105
26 to 281.9093334138
28 to 30 1.9135664414
30 to 321.9173761529
32 to 34 1.9208257688
34 to 361.9239662228
36 to 38 zeros1.9268390530
38 to 40 1.9294785049
40 to 42 1.9319130876
42 to 441.9341667441
44 to 461.9362597431
46 to 481.9382093682
48 to 501.9400304552
50 to 521.9417358171
52 to 541.9433365818
54 to 561.9448424643
56 to 581.9462619872
58 to 601.9476026606
60 to 621.9488711300
62 to 641.9500732990
64 to 661.9512144314
66 to 681.9522992368
68 to 701.9533319424
70 to 72 zeros1.9543163543
72 to 741.9552559090
77 to 761.9561537182
76 to 781.9570126063
78 to 801.9578351437
80 to 821.9586236746
82 to 841.9593803423
84 to 861.9601071098
86 to 881.9608057791
88 to 901.9614780074
90 to 921.9621253217
92 to 941.9627491312
94 to 961.9633507390
96 to 981.9639313515
98 to 1001.9644920877
100 to 1021.9650339870
102 to 1041.9655580163