反常积分最重要的函数之伽马函数 | 您所在的位置:网站首页 › 反常积分函数叫什么 › 反常积分最重要的函数之伽马函数 |
伽马函数
常用于概率论的计算,其实凑正态也行 Γ ( α ) = ∫ 0 + ∞ x α − 1 e − x d x \Gamma (\alpha) = \int_{0}^{+ \infty}x ^{\alpha -1}e^{-x}dx Γ(α)=∫0+∞xα−1e−xdx 如: ∫ 0 + ∞ x 5 e − x d x = Γ ( 5 + 1 ) \int_{0}^{+ \infty}x ^{5}e^{-x}dx = \Gamma (5 + 1) ∫0+∞x5e−xdx=Γ(5+1) ∫ 0 + ∞ x e − x d x = Γ ( 1 2 + 1 ) \int_{0}^{+ \infty} \sqrt{x} e^{-x}dx = \Gamma (\frac{1}{2} + 1) ∫0+∞x e−xdx=Γ(21+1) 性质Γ ( α + 1 ) = α Γ ( α ) \Gamma (\alpha + 1) = \alpha \Gamma (\alpha) Γ(α+1)=αΓ(α) Γ ( 1 2 ) = π \Gamma (\frac{1}{2}) = \sqrt{\pi } Γ(21)=π Γ ( n + 1 ) = n ! \Gamma (n + 1) = n ! Γ(n+1)=n! 例题 ∫ 0 + ∞ x 3 e − 2 x d x = 1 16 ∫ 0 + ∞ ( 2 x ) 3 e − 2 x d ( 2 x ) = 1 16 Γ ( 3 + 1 ) = 3 ! 16 = 3 8 \int_{0}^{+ \infty}x ^{3}e^{-2x}dx = \frac{1}{16}\int_{0}^{+ \infty}(2x) ^{3}e^{-2x}d(2x) = \frac{1}{16} \Gamma (3 + 1) = \frac{3!}{16} = \frac{3}{8} ∫0+∞x3e−2xdx=161∫0+∞(2x)3e−2xd(2x)=161Γ(3+1)=163!=83 ∫ 0 + ∞ x 4 e − x 2 d x = 1 2 ∫ 0 + ∞ ( x 2 ) 3 2 e − x 2 d x 2 = 1 2 Γ ( 3 2 + 1 ) = 1 2 3 2 Γ ( 1 2 + 1 ) = 1 2 3 2 1 2 Γ ( 1 2 ) = 1 2 3 2 1 2 π = 3 8 π \int_{0}^{+ \infty}x ^{4}e^{-x^{2}}dx = \frac{1}{2} \int_{0}^{+ \infty}(x ^{2})^{\frac{3}{2}}e^{-x^{2}}dx^{2} = \frac{1}{2}\Gamma (\frac{3}{2} + 1) = \frac{1}{2} \frac{3}{2}\Gamma (\frac{1}{2} + 1) = \frac{1}{2} \frac{3}{2} \frac{1}{2} \Gamma (\frac{1}{2}) = \frac{1}{2} \frac{3}{2} \frac{1}{2}\sqrt{\pi } = \frac{3}{8}\sqrt{\pi } ∫0+∞x4e−x2dx=21∫0+∞(x2)23e−x2dx2=21Γ(23+1)=2123Γ(21+1)=212321Γ(21)=212321π =83π |
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