오일러 베타적분(베타함수)

\(B(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt\)

\(B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}\)

\(B(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta\)

\(\int_0^{\frac{\pi}{2}}\sin^{p}\theta{d\theta}= \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{p}{2}+\frac{1}{2})}{\Gamma(\frac{p}{2}+1)}\)

\(\int_0^{\frac{\pi}{2}}\sin^{2n}\theta{d\theta}= \frac{\sqrt{\pi}\Gamma(n+\frac{1}{2})}{2\Gamma(n+1)}=\frac{\pi}{2}\frac{(\frac{1}{2})_n}{(1)_n}\)

(정리)

\(a,b,a+b \in \mathbb{Q-Z}\) 라 하자. \(B(a,b)\) 는 초월수이다. 즉

\(B(a,b) = \frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}= \int_0^1t^{a-1}(1-t)^{b-1}\,dt\)

는 초월수이다.

lemniscate 곡선의 길이와 타원적분
\(L=2\omega=4\int_0^1\frac{dx}{\sqrt{1-x^4}}=B(1/2,1/4)=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}=5.24\cdots\)

목차