Research Article
An Alpha-Beta Phase Diagram Representation of the Zeros and Properties of the Mittag-Leffler Function
Table 3
Reference values in phase
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(a) Reference value transition points along the line |
| Transition | Alpha |
| 1 to 3 zeros | 1.3395747103 | 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 15 | 1.7244096473 | 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 27 | 1.8106627476 | 27 to 29 | 1.8195027301 | 29 to 31 | 1.8274511290 | 31 to 33 | 1.8346430576 | 33 to 35 | 1.8411868715 | 35 to 37 zeros | 1.8471706026 | 37 to 39 | 1.8526665973 | 39 to 41 | 1.8577349238 | 41 to 43 | 1.8624259162 | 43 to 45 | 1.8667821039 | 45 to 47 | 1.8708396938 | 47 to 49 | 1.8746297224 | 49 to 51 | 1.8781789624 | 51 to 53 | 1.8815106427 | 53 to 55 | 1.8846450251 | 55 to 57 | 1.8875998709 | 57 to 59 | 1.8903908216 | 59 to 61 | 1.8930317103 | 61 to 63 | 1.8955348200 | 63 to 65 | 1.8979110982 | 65 to 67 | 1.9001703362 | 67 to 69 | 1.9023213199 | 69 to 71 zeros | 1.9043719579 | 71 to 73 | 1.9063293896 | 73 to 75 | 1.9082000780 | 75 to 77 | 1.9099898892 | 77 to 79 | 1.9117041604 | 79 to 81 | 1.9133477593 | 81 to 83 | 1.9149251354 | 83 to 85 | 1.9164403642 | 85 to 87 | 1.9178971868 | 87 to 89 | 1.9192990435 | 89 to 91 | 1.9206491039 | 91 to 93 | 1.9219502937 | 93 to 95 | 1.9232053177 | 95 to 97 | 1.9244166806 | 97 to 99 | 1.9255867056 | 99 to 101 | 1.9267175505 | 101 to 103 | 1.9278112225 |
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(b) Reference value transition points along the horizontal line |
| Transition | Alpha |
| 1 to 3 zeros | 1.4221906908 | 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 15 | 1.7813926516 | 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 27 | 1.8526111866 | 27 to 29 | 1.8597618108 | 29 to 31 | 1.8661681768 | 31 to 33 | 1.8719460965 | 33 to 35 | 1.8771879211 | 35 to 37 zeros | 1.8819682945 | 37 to 39 | 1.8863482727 | 39 to 41 | 1.8903783311 | 41 to 43 | 1.8941005978 | 43 to 45 | 1.8975505379 | 45 to 47 | 1.9007582408 | 47 to 49 | 1.9037494173 | 49 to 51 | 1.9065461804 | 51 to 53 | 1.9091676623 | 53 to 55 | 1.9116305080 | 55 to 57 | 1.9139492725 | 57 to 59 | 1.9161367439 | 59 to 61 | 1.9182042070 | 61 to 63 | 1.9201616614 | 63 to 65 | 1.9220180019 | 65 to 67 | 1.9237811690 | 67 to 69 | 1.9254582752 | 69 to 71 zeros | 1.9270557120 | 71 to 73 | 1.9285792400 | 73 to 75 | 1.9300340659 | 75 to 77 | 1.9314249085 | 77 to 79 | 1.9327560555 | 79 to 81 | 1.9340314117 | 81 to 83 | 1.9352545420 | 83 to 85 | 1.9364287076 | 85 to 87 | 1.9375568985 | 87 to 89 | 1.9386418610 | 89 to 91 | 1.9396861230 | 91 to 93 | 1.9406920148 | 93 to 95 | 1.9416616889 | 95 to 97 | 1.9425971365 | 97 to 99 | 1.9435002028 | 99 to 101 | 1.9443725998 | 101 to 103 | 1.9452159190 |
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(c) Reference value transition points along the line |
| Transition | Alpha |
| 2 to 4 zeros | 1.2916629732 | 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 16 | 1.6787594800 | 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 28 | 1.7751811559 | 28 to 30 | 1.7852650145 | 30 to 32 | 1.7943644328 | 32 to 34 | 1.8026245763 | 34 to 36 | 1.8101625260 | 36 to 38 zeros | 1.8170738953 | 38 to 40 | 1.8234376334 | 40 to 42 | 1.8293195759 | 42 to 44 | 1.8347751111 | 44 to 46 | 1.8398512130 | 46 to 48 | 1.8445880105 | 48 to 50 | 1.8490200131 | 50 to 52 | 1.8531770789 | 52 to 54 | 1.8570851871 | 54 to 56 | 1.8607670588 | 56 to 58 | 1.8642426610 | 58 to 60 | 1.8675296189 | 60 to 62 | 1.8706435553 | 62 to 64 | 1.8735983713 | 64 to 66 | 1.8764064818 | 66 to 68 | 1.8790790112 | 68 to 70 | 1.8816259606 | 70 to 72 zeros | 1.8840563472 | 72 to 74 | 1.8863783254 | 77 to 76 | 1.8885992882 | 76 to 78 | 1.8907259561 | 78 to 80 | 1.8927644526 | 80 to 82 | 1.8947203703 | 82 to 84 | 1.8965988280 | 84 to 86 | 1.8984045205 | 86 to 88 | 1.9001417625 | 88 to 90 | 1.9018145263 | 90 to 92 | 1.9034264764 | 92 to 94 | 1.9049809984 | 94 to 96 | 1.9064812261 | 96 to 98 | 1.9079300641 | 98 to 100 | 1.9093302094 | 100 to 102 | 1.9106841692 | 102 to 104 | 1.9119942782 |
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(d) Reference value transition points along the horizontal line |
| Transition | Alpha |
| 2 to 4 zeros | 1.3395747103 | 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 16 | 1.7244096473 | 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 28 | 1.8106627476 | 28 to 30 | 1.8195027301 | 30 to 32 | 1.8274511290 | 32 to 34 | 1.8346430576 | 34 to 36 | 1.8411868715 | 36 to 38 zeros | 1.8471706026 | 38 to 40 | 1.8526665973 | 40 to 42 | 1.8577349238 | 42 to 44 | 1.8624259162 | 44 to 46 | 1.8667821039 | 46 to 48 | 1.8708396938 | 48 to 50 | 1.8746297224 | 50 to 52 | 1.8781789624 | 52 to 54 | 1.8815106427 | 54 to 56 | 1.8846450251 | 56 to 58 | 1.8875998709 | 58 to 60 | 1.8903908216 | 60 to 62 | 1.8930317103 | 62 to 64 | 1.8955348200 | 64 to 66 | 1.8979110982 | 66 to 68 | 1.9001703362 | 68 to 70 | 1.9023213199 | 70 to 72 zeros | 1.9043719579 | 72 to 74 | 1.9063293896 | 77 to 76 | 1.9082000780 | 76 to 78 | 1.9099898892 | 78 to 80 | 1.9117041604 | 80 to 82 | 1.9133477593 | 82 to 84 | 1.9149251354 | 84 to 86 | 1.9164403642 | 86 to 88 | 1.9178971868 | 88 to 90 | 1.9192990435 | 90 to 92 | 1.9206491039 | 92 to 94 | 1.9219502937 | 94 to 96 | 1.9232053177 | 96 to 98 | 1.9244166806 | 98 to 100 | 1.9255867056 | 100 to 102 | 1.9267175505 | 102 to 104 | 1.9278112225 |
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(e) Reference value transition points along the horizontal line |
| Transition | Alpha |
| 2 to 4 zeros | 1.7100323859 | 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 16 | 1.8684122443 | 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 28 | 1.9093334138 | 28 to 30 | 1.9135664414 | 30 to 32 | 1.9173761529 | 32 to 34 | 1.9208257688 | 34 to 36 | 1.9239662228 | 36 to 38 zeros | 1.9268390530 | 38 to 40 | 1.9294785049 | 40 to 42 | 1.9319130876 | 42 to 44 | 1.9341667441 | 44 to 46 | 1.9362597431 | 46 to 48 | 1.9382093682 | 48 to 50 | 1.9400304552 | 50 to 52 | 1.9417358171 | 52 to 54 | 1.9433365818 | 54 to 56 | 1.9448424643 | 56 to 58 | 1.9462619872 | 58 to 60 | 1.9476026606 | 60 to 62 | 1.9488711300 | 62 to 64 | 1.9500732990 | 64 to 66 | 1.9512144314 | 66 to 68 | 1.9522992368 | 68 to 70 | 1.9533319424 | 70 to 72 zeros | 1.9543163543 | 72 to 74 | 1.9552559090 | 77 to 76 | 1.9561537182 | 76 to 78 | 1.9570126063 | 78 to 80 | 1.9578351437 | 80 to 82 | 1.9586236746 | 82 to 84 | 1.9593803423 | 84 to 86 | 1.9601071098 | 86 to 88 | 1.9608057791 | 88 to 90 | 1.9614780074 | 90 to 92 | 1.9621253217 | 92 to 94 | 1.9627491312 | 94 to 96 | 1.9633507390 | 96 to 98 | 1.9639313515 | 98 to 100 | 1.9644920877 | 100 to 102 | 1.9650339870 | 102 to 104 | 1.9655580163 |
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