Seznam integralov Gaussovih funkcij

Seznam integralov Gaussovih funkcij vsebuje integrale Gaussovih funkcij.

V pregledu pomeni $\phi(x) = \tfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}$ funkcijo gostote verjetnosti za normalno porazdelitev in $\textstyle \Phi(x) = \int_{-\infty}^x \phi(t)dt = \frac12\big(1 + \operatorname{erf}\big(\frac{x}{\sqrt{2}}\big)\big)$ je pripadajoča zbirna funkcija verjetnosti (kjer je erf funkcija napake).

Nedoločeni integrali

$\int \phi(x) \, dx = \Phi(x) + C$

$\int x \phi(x) \, dx = -\phi(x) + C$

$\int x^2 \phi(x) \, dx = \Phi(x) - x\phi(x) + C$

$\int x^{2k+1} \phi(x) \, dx = -\phi(x) \sum_{j=0}^k \frac{(2k)!!}{(2j)!!}x^{2j} + C$

$\int x^{2k+2} \phi(x) \, dx = -\phi(x)\sum_{j=0}^k\frac{(2k+1)!!}{(2j+1)!!}x^{2j+1} + (2k+1)!!\,\Phi(x) + C$

(v teh integralih je n!! dvojna fakulteta: za parne vrednosti n je to zmnožek vseh parnih števil od 2 do n, za neparne n je to zmnožek vseh neparnih števil od 1 do n, dodatno še velja 1=0!! = (−1)!! = 1).

$\int \phi(x)^2 \, dx = \tfrac{1}{2\sqrt{\pi}} \Phi(x\sqrt{2}) + C$

$\int \phi(x)\phi(a + bx) \, dx = \tfrac{1}{t}\phi(a/t)\Phi(tx + ab/t) + C, \quad t = \sqrt{1+b^2}$ : $\int x\phi(a+bx) \, dx = -\tfrac{1}{b^2}\phi(a+bx) - \tfrac{a}{b^2}\Phi(a+bx) + C$

$\int x^2\phi(a+bx) \, dx = \tfrac{a^2+1}{b^3}\Phi(a+bx) + \frac{a-bx}{b^3}\phi(a+bx) + C$

$\int \phi(a+bx)^n \, dx = \frac{(2\pi)^{-(n-1)/2}}{b\sqrt{n}} \Phi\big(\sqrt{n}(a+bx)\big) + C$

$\int \Phi(a+bx) \, dx = \tfrac{1}{b}(a+bx)\Phi(a+bx) + \tfrac{1}{b}\phi(a+bx) + C$

$\int x\Phi(a+bx) \, dx = \tfrac{1}{2b^2}\big((b^2x^2 - a^2 - 1)\Phi(a+bx) + (bx-a)\phi(a+bx)\big) + C$

$\int x^2\Phi(a+bx) \, dx = \tfrac{1}{3b^3}\big((b^3x^3 + a^3 + 3a)\Phi(a+bx) + (b^2x^2-abx+a^2+2)\phi(a+bx)\big) + C$

$\int x^n \Phi(x) \, dx = \tfrac{1}{n+1}\Big( (x^{n+1}-nx^{n-1})\Phi(x) + x^n\phi(x) + n(n-1)\int x^{n-2}\Phi(x)\,dx \Big) + C$

$\int x\phi(x)\Phi(a+bx) \, dx = \tfrac{b}{t}\phi(a/t)\Phi(xt + ab/t) - \phi(x)\Phi(a+bx) + C, \quad t = \sqrt{1+b^2}$

$\int \Phi(x)^2 \, dx = x \Phi(x)^2 + 2\Phi(x)\phi(x) - \tfrac{1}{\sqrt{\pi}}\Phi(x\sqrt{2}) + C$

$\int e^{cx}\phi(bx)^n \, dx = \frac{1}{b\sqrt{n(2\pi)^{n-1}}}e^{c^2/(2nb^2)}\Phi(bx\sqrt{n} - \tfrac{c}{b\sqrt{n}}) + C, \quad b\ne0, n>0$

Določeni integrali

$\begin{align} & \int_{-\infty}^\infty x^2\phi(x)^n \, \, dx = \Big(n^{3/2}(2\pi)^{(n-1)/2}\Big)^{-1} \\ & \int_{-\infty}^0 \phi(ax)\Phi(bx)dx = (2\pi a)^{-1}\arctan(a/b) \\ & \int_0^{\infty} \phi(ax)\Phi(bx) \, dx = (2\pi a)^{-1}\big(\tfrac{\pi}{2} - \arctan(b/a)\big) \\ & \int_0^\infty x\phi(x)\Phi(bx) \, dx = \frac{1}{2\sqrt{2\pi}} \bigg( 1 + \frac{b}{\sqrt{1+b^2}} \bigg) \\ & \int_0^\infty x^2\phi(x)\Phi(bx) \, dx = \frac14 + \frac{1}{2\pi}\bigg( \frac{b}{1+b^2} + \arctan b \bigg) \\ & \int x \phi(x)^2\Phi(x) \, dx = \frac{1}{4\pi\sqrt{3}} \\ & \int_0^\infty \Phi(bx)^2 \phi(x) \, dx = (2\pi)^{-1}\big( \arctan b + \arctan \sqrt{1+2b^2} \big) \\ & \int_{-\infty}^\infty \Phi(bx)^2 \phi(x) \, dx = \pi^{-1}\arctan \sqrt{1+2b^2} \\ & \int_{-\infty}^\infty x\phi(x)\Phi(bx) \, dx = \int_{-\infty}^\infty x\phi(x)\Phi(bx)^2 \, dx = \frac{b}{\sqrt{2\pi(1+b^2)}} \\ & \int_{-\infty}^\infty \Phi(a+bx)\phi(x) \, dx = \Phi\big(a/\sqrt{1+b^2}\big) \\ & \int_{-\infty}^\infty x\Phi(a+bx)\phi(x) \, dx = (b/t)\phi(a/t), \quad t = \sqrt{1+b^2} \\ & \int_0^\infty x\Phi(a+bx)\phi(x) \, dx = (b/t)\phi(a/t)\Phi(-ab/t) + (2\pi)^{-1/2}\Phi(a), \quad t = \sqrt{1+b^2} \\ & \int_{-\infty}^\infty \ln(x^2) \tfrac{1}{\sigma}\phi\big(\tfrac{x}{\sigma}\big) \, dx = \ln(\sigma^2) - \gamma - \ln 2 \approx \ln(\sigma^2) - 1.27036 \end{align}$

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