∫√(3 - x^2)dx⋍(x/2)√(3 - x^2)+(3/2) ArcSin[x/√3] | ||||||||
② | √(q^2-x^2)⋍ | √q^2(1-k^2・x^2)⋍ | qE | (第2種) | ||||
⋍(qπ/2)F[-1/2,1/2,1;k^2]⋍(1-s)E | ||||||||
②9.8∫√(2k+1-x^2)dx | ⋍9.8{φ+(1/2)Sin2φ} | |||||||
∫√(3 - x^2)dxの | 積分は計算できるx=1を代入) | |||||||
(x/2)√(3 - x^2)+ | (3/2) ArcSin[x/√3] | ⋍ | 9.8Σ | ; | 文献値 | |||
0.707107 | + | 0.9232196 | ⋍ | 1.630326 | ⋍ | 15.9772 | ; | 14.13472 |
(x/2)√(5 - x^2)+ | (5/2) ArcSin[x/√5] | |||||||
1 | + | 1.1591190 | ⋍ | 2.159119 | ⋍ | 21.15937 | ; | 21.02203 |
(x/2)√(7 - x^2)+ | (7/2) ArcSin[x/√7] | |||||||
1.224745 | + | 1.3565884 | ⋍ | 2.581333 | ⋍ | 25.29707 | ; | 25.01085 |
(x/2)√(9 - x^2)+ | (9/2) ArcSin[x/√9] | |||||||
1.414214 | + | 1.5292661 | ⋍ | 2.943480 | ⋍ | 28.8461 | ; | 30.42487 |
(x/2)√(11 - x^2)+ | (11/2) ArcSin[x/√11] | |||||||
1.581139 | + | 1.6845255 | ⋍ | 3.265664 | ⋍ | 32.00351 | ; | 32.93506 |
(x/2)√(13 - x^2)+ | (13/2) ArcSin[x/√13] | |||||||
1.732051 | + | 1.8267269 | ⋍ | 3.558778 | ⋍ | 34.87602 | ; | |
(x/2)√(15 - x^2)+ | (15/2) ArcSin[x/√15] | |||||||
1.936492 | + | 1.9586806 | ⋍ | 3.895172 | ⋍ | 38.17269 | ; | 37.58617 |
(x/2)√(17 - x^2)+ | (17/2) ArcSin[x/√17] | |||||||
2 | + | 2.0823186 | ⋍ | 4.082319 | ⋍ | 40.00672 | ; | |
(x/2)√(19 - x^2)+ | (19/2) ArcSin[x/√19] | |||||||
2.12132 | + | 2.1990350 | ⋍ | 4.320355 | ⋍ | 42.33948 | ; | 43.32707 |
(x/2)√(21 - x^2)+ | (21/2) ArcSin[x/√21] | |||||||
2.236068 | + | 2.3098738 | ⋍ | 4.545942 | ⋍ | 44.55023 | ; | |
(x/2)√(23 - x^2)+ | (23/2) ArcSin[x/√23] | |||||||
2.345208 | + | 2.4156410 | ⋍ | 4.760849 | ⋍ | 46.65632 | ; | 48.00515 |
(x/2)√(25 - x^2)+ | (25/2) ArcSin[x/√25] | |||||||
2.44949 | + | 2.5169740 | ⋍ | 4.966464 | ⋍ | 48.67134 | ; | |
(x/2)√(27 - x^2)+ | (27/2) ArcSin[x/√27] | |||||||
2.54951 | + | 2.6143871 | ⋍ | 5.163897 | ⋍ | 50.60619 | ; | 49.77383 |
(x/2)√(29 - x^2)+ | (29/2) ArcSin[x/√29] | |||||||
2.645751 | + | 2.7083022 | ⋍ | 5.354053 | ⋍ | 52.46972 | ; | |
(x/2)√(31 - x^2)+ | (31/2) ArcSin[x/√31] | |||||||
2.738613 | + | 2.7990708 | ⋍ | 5.537684 | ⋍ | 54.2693 | ; | |
(x/2)√(33 - x^2)+ | (33/2) ArcSin[x/√33] | |||||||
2.828427 | + | 2.8869893 | ⋍ | 5.715416 | ⋍ | 56.01108 | ; | 56.446 |
(x/2)√(39 - x^2)+ | (39/2) ArcSin[x/√39] | |||||||
3.082207 | + | 3.1359994 | ⋍ | 6.218206 | ⋍ | 60.93842 | ; | 59.347 |
(x/2)√(45 - x^2)+ | (45/2) ArcSin[x/√45] | |||||||
3.316625 | + | 3.3666505 | ⋍ | 6.683275 | ⋍ | 65.4961 | ; | 65.112 |
リーマン・ゼータの零点を考察した。 | |||||
①11で割リ、連分数が1となるように(s/11)を加えると√2k+1が導かれる。 | |||||
②√2k+1でわると比例定数9.8が得られる。 | |||||
③9.8√2k+1+Sin[x](特解) | |||||
④別計算 | |||||
⑤その積分をMathematica | |||||
⑥ルジャンドルの第2種積分で表現してプランクの公式とした。 |
⓪ | 11(√2k+1-s/11) | (mod 11 | ||
k | 文献値 | 文/11 | +(s/11) | √Q⋍√2k+1 |
1 | 14.13472 | 1.2849745 | 0.45454545 | 1.73952 |
2 | 21.02203 | 1.9110936 | 0.36363636 | 2.27473 |
3 | 25.01085 | 2.2737136 | 0.36363636 | 2.63735 |
4 | 3 | |||
5 | 30.42487 | 2.7658973 | 0.72727273 | 3.49317 |
6 | 32.93506 | 2.9940964 | 0.63636364 | 3.63046 |
7 | 37.58617 | 3.4169245 | 0.45454545 | 3.87147 |
8 | 40.91871 | 3.7198827 | 0.27272727 | 3.99261 |
9 | 43.32707 | 3.9388245 | 0.45454545 | 4.39337 |
10 | 4.58257569 | |||
11 | 48.00515 | 4.3641045 | 0.45454545 | 4.81865 |
12 | 49.77383 | 4.5248936 | 0.36363636 | 4.88853 |
13 | 52.97 | 4.8154545 | 0.45454545 | 5.27 |
14 | 56.446 | 5.1314545 | 0.45454545 | 5.586 |
15 | 59.347 | 5.3951818 | 0.18181818 | 5.577 |
16 | 60.831 | 5.5300909 | 0.09090909 | 5.621 |
17 | 5.91607978 | |||
18 | 65.112 | 5.9192727 | 0.27272727 | 6.192 |
29 | 7.61577311 | |||
30 | 75.8 | 6.8909091 | 0.90909091 | 7.8 |