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User:Tevatron

我爱勒让德函数

(1-x^2)\frac{\mathrm{d}^2\Theta}{\mathrm{d}x^22x\frac{\mathrm{d}\Theta}{\mathrm{d}x}+l(l+1)\Theta=0}-

\mathrm{P}_6(x)=\frac{1}{16}(231x^6-315x^4+105x^2-5)=\frac{1}{512}(231\cos6\theta+126\cos4\theta+105\cos2\theta+50)

[编辑] 高斯积分

I=\int^\infty_0\mathrm{e}^{-x^2}\,\mathrm{d}x

\left[\int_{-\infty}^\infty\mathrm{e}^{-x^2}\,\mathrm{d}x\right]^2=\int\!\!\int\mathrm{e}^{-(x^2+y^2)}\,\mathrm{d}x\,\mathrm{d}y=\int\!\!\int\mathrm{e}^{-r^2}r\,\mathrm{d}r\,\mathrm{d}\theta=\pi

\left[\int_{-\infty}^\infty\mathrm{e}^{-x^2}\,\mathrm{d}x\right]^2=\int\int\mathrm{e}^{-(x^2+y^2)}\,\mathrm{d}x\,\mathrm{d}y=\int\int\mathrm{e}^{-r^2}r\,\mathrm{d}r\,\mathrm{d}\theta=\pi

I=\frac{\sqrt{\pi}}{2}


[编辑] Γ函数

定义：

\Gamma(x)=\int^\infty_0\mathrm{e}^{-t}\,t^{x-1}\,\mathrm{d}t

递推关系：

Γ(x + 1) = xΓ(x)

常用泛函关系：

\Gamma(1-x)\Gamma(x)=\frac{\pi}{\sin\pi{}x} \Gamma(\frac12+x)\Gamma(\frac12-x)=\frac{\pi}{\cos\pi{}x}

n为正整数时：

Γ(n + 1) = n! \Gamma(n+\frac12)=\frac{\sqrt\pi}{2^n}(2n-1)!! \Gamma(\frac12-n)=(-1)^n\frac{2^n\sqrt\pi}{(2n-1)!!}

特殊值：

Γ(2) = Γ(1) = 1 \Gamma(\frac12)=\sqrt\pi \Gamma(-\frac12)=-2\sqrt\pi

[编辑] 勒让德函数

勒让德方程：

(1-x^2)\frac{\mathrm{d}^2\Theta}{\mathrm{d}x^22x\frac{\mathrm{d}\Theta}{\mathrm{d}x}+l(l+1)\Theta=0}-

勒让德多项式：

P0(x) = 1 P0(cosθ) = 1 P1(x) = x P1(cosθ) = cosθ \mathrm{P}_2(x)=\frac12(3x^2-1) \mathrm{P}_2(\cos\theta)=\frac14(3\cos2\theta+1) \mathrm{P}_3(x)=\frac12(5x^3-3x) \mathrm{P}_3(\cos\theta)=\frac18(5\cos3\theta+3\cos\theta) \mathrm{P}_4(x)=\frac18(35x^4-30x^2+3) \mathrm{P}_4(\cos\theta)=\frac{1}{64}(35\cos4\theta+20\cos2\theta+9) \mathrm{P}_5(x)=\frac18(63x^5-70x^3+15x) \mathrm{P}_5(\cos\theta)=\frac{1}{128}(\cos5\theta+35\cos3\theta+30\cos\theta) \mathrm{P}_6(x)=\frac{1}{16}(231x^6-315x^4+105x^2-5) \mathrm{P}_6(\cos\theta)=\frac{1}{512}(231\cos6\theta+126\cos4\theta+105\cos2\theta+50)

来自“http://www.wikilib.com.cn/wiki/User:Tevatron”