| ①f(s)⋍f(1-s)or f(θ)⋍f(π-θ)成立する | ||||||
| ∴ξ(s)⋍ξ(1-s) | ||||||
| ②1/(Γs･Γ1-s)⋍1/B[s,1-s]⋍(1/π)Sin(sπ)相反公式 | ||||||
| π/(Γs･Γ1-s)⋍π/B[s,1-s]⋍Sin(sπ) | ||||||
| ∴{Sin(sπ)｝^2⋍{π/(Γs･Γ1-s)}^2⋍π/B[s,1-s]{}^2⋍(-1/4)[1/k-1/k+1] | ||||||
| ③abc予想より、ξ(s)ξ(1-s)⋍ξ(s)+ξ(1-s) | ||||||
| (ﾌｧｲﾝﾏﾝ) | 定積分の分母を式変形 | |||||
| [αA+（1-α）B]^2=［(A-B)α+B］^2= | ||||||
| (A-B)^2⋆[α+k]＾2 | ∵B/(A-B)≅k | |||||
| ∴1/AB=1/(A-B)^2⋆∫dα/[α+k]^2= | ||||||
| 1/(A-B)^2⋆[1/(1+k)-1/k] | ||||||
| ③Ａ=e^iθ，B=e^-iθとおくと、ﾌｧｲﾝﾏﾝ･ﾊﾟﾗﾒｰﾀ | ||||||
| 1/AB=1/(A-B)^2⋆[1/k-1/k+1] | ||||||
| ∴-4y^2=(2iSinθ)^2=[1/k-1/k+1]=-[1/(s-1/2)-1/(s+1/2)] | ||||||
| 両辺を積分して | ||||||
| ∫Sin^2(t)dt=(-1/4)LN(s-1)/(s+1)（複比,非調和)=d(光錐上の距離p86定理7.1) | ||||||
| ～(t/2 - 1/4 Sin[2 t])≒t/2 (∵Sinθ≒0) | ||||||
| ④零点⋍11⒤{√(2m+1)-s/11} | ||||||
| fがu方向の偏微係数といえばuは偏微分作用素u⋍u(a/ax) | ||||
| ∴u=(a,a),v=(-1/2,0)とおけばよい | 松本p86 | |||
| the minimizer f is 2Δf-|▽f|^2+R=const.(ﾍﾟﾚﾙﾏﾝ､p5) | ||||
| the function f on a closed manfiled M (ﾍﾟﾚﾙﾏﾝ､p8) | ||||
| u=(1,1),v=(-1/2,0)とおく | ||||
| f=u+⒤v ∴⒤ｆ=ｰvu+⒤u | ｻﾞｷﾞｬｰ表記 | |||
| ⃣②γ={{1,1}､{0,1}} | [a,b,c] | |||
| ∴γ^2={{1,2},{0,1}} | [1,-1,3.25] | |||
| ③γ^2⒤v=(-1/2,0) | [1,-1,5.25] | |||
| γ^2･⒤u=(3,1) | … | |||
| γ^4･⒤u=(5,1) | [1,-1,(2m+1).25] | |||
| … | [1,-1,q.25] | |||
| γ^2m･⒤u=(2m+1,1) | ∴ｘ⋍1/2∓⒤√Q⋍√(2m+1) | |||
| ∴√γ^2m･⒤u=√(2m+1,1)⋍√(2m+1) | x⋍1/2∓⒤1.73205… | |||
| 上式を(□-m＾2)φ⋍0(ｸﾗｲﾝ･ｺﾞﾙﾄﾞﾝ方程式) | ε7⋍15/2hν | |||
| ※零点⋍11(√(２m+1)-s/11) | ε6⋍13/2hν | |||
| ④(ﾘｰﾏﾝ)零点⋍円の方程式の解 | ε5⋍11/2hν | |||
| x＾2-ｘ＋Q．25⋍(x-1/2)^2+Q=0 | ε4⋍9/2hν | |||
| ∴ｘ⋍1/2∓⒤√Q⋍√(2m+1) | ε3⋍7/2hν | |||
| ⑤¦u|⋍√２なので | ε2⋍5/2hν | |||
| √(2m+1)/√2⋍√(m+1/2)∼ε | ε1⋍3/2hν | |||
| ∴εn⋍(n+1/2)hν | ε0⋍1/2hν | |||
| (p63)調和振動の確率関数 | ||||
| ∴εn⋍(n+1/2)hν | ||||
| Γ(q.25) | ⋍δ+θ⋍δ+q.25 | |||
| s⋍q.25 | Γ(q.25) | δΓ(s) | δ/δ⋍θ | k＾2 |
| 0.25 | 3.62561 | 連分数 | ||
| 1.25 | 0.906402 | -2.719208 | ⋍q.25⋍ | |
| 2.25 | 1.133 | 0.226598 | 6.2500993 | (5/２)^2 |
| 3.25 | 2.54926 | 1.41626 | 4.0499838 | ⋍(9/√20)^2 |
| 4.25 | 8.28509 | 5.73583 | 4.69444 | ⋍(13/6)^2 |
| 5.25 | 35.2116 | 26.92651 | 5.5576976 | ⋍50/9 |
| 6.25 | 184.861 | 149.6494 | 6.4852849 | ⋍q.25 |
| 7.25 | 1155.38 | 970.519 | 7.4404829 | ⋍q.25 |
| 8.25 | 8376.51 | 7221.13 | 8.4099982 | ⋍q.25 |
| 9.25 | 69106.2 | 60729.69 | 9.3879419 | ⋍q.25 |
| 10.25 | 639233 | 570126.8 | 10.371203 | ⋍q.25 |
| 11.25 | 6552134 | 5912901 | 11.358109 | ⋍q.25 |
| 12.25 | 73711509 | 67159375 | 12.347561 | ⋍q.25 |
| 13.25 | 902965985 | 829254476 | 13.33889 | ⋍q.25 |
| 14.25 | 1.196E+10 | 1.106E+10 | 14.331632 | ⋍q.25 |
| 15.25 | 1.705E+11 | 1.585E+11 | 15.325472 | ⋍q.25 |
| 16.25 | 2.6E+12 | 2.43E+12 | 16.320175 | ⋍q.25 |
| ∵ΓsΓ1-s⋍B(s,1-s)⋍π/Sin(sπ) | ∴B(s,1-s)⋍δΓs+δΓ1-s | ||||||
| δΓs | ⋍p/q | Γ1-s⋍π/Γs | δΓ1-s | ⋍q/p | ⋍p/q+q/p | s | 1-s |
| 162.42 | ⋍110395/64 | 0.010912655 | 0.0061566 | ⋍64/110395 | 162.424 | 6.5 | -5.5 |
| 29.53 | ⋍945/32 | 0.060019601 | 0.0338616 | ⋍32/945 | 29.5643 | 5.5 | -4.5 |
| 6.5623 | ⋍105/16 | 0.270088206 | 0.152377 | ⋍16/105 | 6.71471 | 4.5 | -3.5 |
| 1.875 | ⋍15/8 | 0.94530872 | 0.5333194 | ⋍8/15 | 2.40827 | 3.5 | -2.5 |
| 0.75 | ⋍3/4 | 2.363271799 | 1.3332986 | ⋍４/3 | 2.08328 | 2.5 | -1.5 |
| ⋍√π/2 | 3.544907698 | ⋍2√π | 2.5√π | 1.5 | -0.5 | ||
| ⋍√π | 1.772453849 | ⋍√π | ⋍π | 0.5 | 0.5 | ||