式展開: x^3・exp(-α x^2)の広義積分[0,∞]

\begin{array}{rcl} \int_{0}^{\infty} x^{3} e^{- α x^{2}} d x \\ = & \int_{0}^{\infty} {(\frac{t}{α})}^{2} e^{- t} \frac{1}{2 \sqrt{α t}} d t \\ \dots t = α x^{2}, x = 0 \to t = 0, x = \infty \to t = \infty, \\ \dots x = \sqrt{\frac{t}{α}} (積 分 範 囲 か ら x \geq 0 と す る (?)) \\ \dots x^{3} = {(\sqrt{\frac{t}{α}})}^{3} = {(\frac{t}{α})}^{\frac{3}{2}} \\ \dots \frac{d t}{d x} = 2 α x, d x = \frac{1}{2 α x} d t = \frac{1}{2 α \sqrt{\frac{t}{α}}} d t = \frac{1}{2 \sqrt{α t}} d t \\ = & \int_{0}^{\infty} {(\frac{t}{α})}^{\frac{3}{2}} e^{- t} \frac{1}{2 \sqrt{α t}} d t \\ = & \frac{1}{2 α^{\frac{3}{2}} \sqrt{α}} \int_{0}^{\infty} \frac{t^{\frac{3}{2}}}{\sqrt{t}} e^{- t} d t \\ = & \frac{1}{2 α^{2}} \int_{0}^{\infty} t e^{- t} d t \\ = & \frac{1}{2 α^{2}} \int_{0}^{\infty} t^{2 - 1} e^{- t} d t \\ = & \frac{1}{2 α^{2}} Γ (2) \dots Γ (z) = \int_{0}^{\infty} t^{z - 1} e^{- t} d t (Re (z) > 0) \\ = & \frac{1}{2 α^{2}} (2 - 1)! \dots Γ (n) = (n - 1)! (n は 自 然 数) \\ = & \frac{1}{2 α^{2}} \cdot 1 \\ = & \frac{1}{2 α^{2}} \end{array}

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