08/01/03 18:33:23
>13,17
I_(2m) = {(2m-1)!!/(2m)!!}(π/2),
I_(2m+1) = {(2m)!!/(2m+1)!!},
I_(n-1)*I_n = π/(2n),
I_(2m)/I(2m-1) = {C[2m,m]/(4^m)}^2 *mπ = exp(-1/(4m) + O(1/m^3)),
I(2m+1)/I(2m) = {(4^m)/C[2m,m]}^2 *2/(π(2m+1)) = exp(-1/(4m) -O(1/m^2)),
より
I(n)/I(n-1) ～ exp(-1/(2n) + O(1/n^2)),
これをn乗して
(与式) → exp(-1/2) = 1/√e,
〔等式064〕
C[2m,m] = (4^m)/√(mπ) * exp(-1/8m + O(1/m^3)) ～ (4^m)/√(mπ) *(1 - 1/(8m) + …),
ｽﾚﾘﾝｸ(math板:064番)
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