Γ(x):=∫∞0e-ttx-1dt,x＞0为gamma函数.设f(x):=logΓ(x)+logΓ(1-x),x∈Q(0,1/2].证明如果存在有理数y0∈Q(0,1/2],使得f(y0)=logΓ(y0)+logΓ(1-y0)∈-Q,则集合{eαπ｜α∈-Q}中恰好有一个代数数,即e-f(y0)π,且e-f(y0)π=sinπy0.%The gamma function is defined as Γ(x):=∫∞0e-ttx-1dt,x ＞ 0.Let∫(x):=logΓ(x) + logΓ(1-x),x ∈ Q(0,1/2] · It is shown that if there exists a rational number y0 ∈ Q(0,1/2] such that f(y0) =logΓ(y0) + logΓ(1-y0) ∈-Q,then there is precisely one algebraic element in the set { eαπ ｜ α ∈-Q},which is e-f(y0) π and e-f(y0) π =sinπy0.
展开▼
