数学 vs エロ

[0001 １３２人目の素数さん 2023/09/29(金) 11:02:14.23 ID:wHXbguzF]

数学はエロに勝てるのか？！

[0002 １３２人目の素数さん 2023/09/29(金) 11:39:26.82 ID:+3oqnmCd]

f: Z → Kに対して、⊿fを

⊿f(x) = f(x + 1) - f(x)
⊿^(n+1)f(x) = ⊿(⊿f)(x)

で定めると

⊿^nf(x) = Σ[k=0, n] (-1)^(n - k) C(n, k) f(x + k)

f(x + n) = Σ[k=0, n] C(n, k)⊿^kf(x)

H = (C(n, k))_{n,k} とおくと、
H^(-1) = ((-1)^(n-k)C(n, n-k))_{n,k}

[0003 １３２人目の素数さん 2023/09/29(金) 12:24:56.05 ID:+3oqnmCd]

⊿f = 0
f(x + 1) = f(x)
e.g. f(x) = exp(2πix)

⊿f = f
f(x + 1) = 2f(x)
e.g. f(x) = exp(log(2)x) = 2^x

⊿f = -f
f(x + 1) = 0
f(x) = 0

⊿f = λf
f(x + 1) = (1 + λ)f(x)
e.g. f(x) = (1 + λ)^x

⊿^2f = f
f(x + 2) - 2f(x + 1)= 0
f(x + 1) = 2f(x)
(⊿ + I)(⊿ - I)f = 0