半径1のn次元球の体積

etc math

Vol(B_n(1)) = \int_{x_1^2 + ... + x_n^2 \leq 1} dx_1 ... dx_n
= \int_{-1}^{1} dx_n \int_{x_1^2 + ... + x_{n-1}^2 \leq 1 - x_n^2} dx_1 ... dx_{n-1}
= Vol(B_{n-1}(1)) \int_{-1}^{1} (\sqrt{1 - x_n^2})^{n-1} dx_n
= 2 Vol(B_{n-1}(1)) \int_{0}^{1} (1-t)^{\frac{n-1}{2}} \frac{dt}{2\sqrt{t}}
= Vol(B_{n-1}(1)) B(\frac{1}{2},\frac{n+1}{2})
= Vol(B_{n-1}(1)) \frac{\Gamma(\frac{1}{2})\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n+2}{2})}
= Vol(B_1(1)) \frac{\Gamma(\frac{1}{2})^{n-1} \Gamma(\frac{3}{2})}{\Gamma(\frac{n+2}{2})}
= \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}
漸化式に落とせる点、好感度高し。ベータ関数とガンマ関数大活躍。
何回か書いた気がするが、未だに微積の単位は無い。ましてや実解析や複素解析なんて。