目次
ガンマ関数
ガンマ関数は以下のように定義する.
$$\Gamma(z)=\displaystyle\int_{0}^{\infty} e^{-t} t^{z-1} dt$$
\(\Gamma(n+1) = n\Gamma(n)\)
\(\Gamma(n+1) \)\(= \displaystyle\int_{0}^{\infty} e^{-x} x^{n} dx=\biggl[-e^{-x} x^n\biggr]_{0}^{\infty}+\displaystyle\int_{0}^{\infty} nx^{n-1} e^x dx\)\(= nΓ(n) \)
\(\Gamma(n+1) = n!\)
\(\Gamma(n+1)\)\(=n\Gamma(n)=n(n-1)\Gamma(n-1)=\cdots=n!\Gamma(1)\)\(n!\)
※\(\Gamma(1)=1\) は定義の式から.
\( \Gamma(\frac{1}{2}) = \sqrt{\pi} \)
\(t=x^2\).
\(\Gamma(\displaystyle\frac{1}{2})=\displaystyle\int_0^{\infty}t^{-\frac{1}{2}}e^{-t}dt=\displaystyle\int_0^{\infty}x^{-1}e^{-x^2}2xdx\)
$=2\displaystyle\int_0^{\infty}e^{-x^2}dx=\sqrt{\pi}$
ベータ関数
\(B(p,q) = \displaystyle\frac{\Gamma(p) \Gamma(q)}{ \Gamma(p+q)} \)
でベータ関数を定義する.
\(B(p,q)= \displaystyle\int_{0}^{1} x^{p-1}(1-x)^{q-1} dx\)
例
その1
\(\displaystyle\int_{0}^{1} \displaystyle\frac{x}{\sqrt{1-x}} dx=\displaystyle\int_{0}^{1} x^{2-1} (1-x)^{\frac{1}{2}-1} dx\)
\(=B\biggl(2 , \displaystyle\frac{1}{2}\biggr)= \displaystyle\frac {\Gamma(2) \Gamma\biggl(\displaystyle\frac{1}{2}\biggr)}{\Gamma\biggl(\displaystyle\frac{5}{2}\biggr)}= \displaystyle\frac{1!\sqrt{\pi}}{\displaystyle\frac{3}{2}\cdot \displaystyle\frac{1}{2}\sqrt{\pi}}=\)\(\displaystyle\frac{4}{3}\)
その2
\( t=\sin^2 \theta \)とおく.
\(\displaystyle\int_{0}^{\frac{\pi}{2}} \sin ^3\theta \cos ^4 \theta d\theta=\displaystyle\frac{1}{2} \int_{0}^{1} t (1-t)^{\frac{3}{2}} dt\)
\(=\displaystyle\frac{1}{2}B \biggl(2,\displaystyle\frac{5}{2}\biggr)=\displaystyle\frac{1}{2}\displaystyle\frac {\Gamma(2) \Gamma\biggl(\displaystyle\frac{5}{2}\biggr)}{\Gamma\biggl(\displaystyle\frac{9}{2}\biggr)}=\displaystyle\frac{1! \Gamma\biggl(\displaystyle\frac{5}{2}\biggr)}{2\cdot \displaystyle\frac{7}{2}\cdot \displaystyle\frac{5}{2}\Gamma\biggl(\displaystyle\frac{5}{2}\biggr)}\)
\(=\displaystyle\frac{2}{35}\)
