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目次
ガンマ関数 倍数公式
\(\Gamma(2z)=\displaystyle\frac{2^{2z-1}}{\sqrt{\pi}}\Gamma(z)\Gamma(z+\displaystyle\frac{1}{2})\)
証明
変形
\(\Gamma(2z)=\displaystyle\frac{\Gamma(z)\Gamma(z)}{B(z,z)}\)
\(=\displaystyle\frac{\Gamma(z)}{B(z,z)}\cdot \displaystyle\frac{B(z,\frac{1}{2})\Gamma(z+\frac{1}{2})}{\Gamma(\frac{1}{2})}\)
\(=\displaystyle\frac{1}{\sqrt{\pi}}\cdot \displaystyle\frac{B(z,\frac{1}{2})}{B(z,z)}\cdot \Gamma(z)\Gamma(z+\frac{1}{2})\)
ベータ関数計算
\(B(z,z)=\displaystyle\int_{0}^{1} t^{z-1} (1-t)^{z-1} dt\)
\(=\displaystyle\int_{\frac{\pi}{2}}^{0} \cos^{2z-2} \theta \sin^{2z-2} \theta (-2\cos\theta\sin\theta)d\theta\) \(\cdots\) \(t=\cos^2\theta\)とした
\(=2\displaystyle\int_{0}^{\frac{\pi}{2}}\biggl(\displaystyle\frac{\sin 2\theta}{2}\biggr)^{2z-1} d\theta\) ※半角など
\(=2^{2-2z}\displaystyle\int_{0}^{\pi}\sin^{2z-1} k \displaystyle\frac{dk}{2}\) ※\(k=2\theta\)
\(=2^{1-2z}\cdot 2\displaystyle\int_{0}^{\frac{\pi}{2}}\sin^{2z-1} k dk\)
\(=2^{1-2z} B(z , \displaystyle\frac{1}{2})\)
よって \(\displaystyle\frac{B(z,\frac{1}{2})}{B(z,z)}=2^{2z-1}\)
結果
\(\Gamma(2z)=\displaystyle\frac{2^{2z-1}}{\sqrt{\pi}}\Gamma(z)\Gamma(z+\displaystyle\frac{1}{2})\)
