查看“︁道森积分”︁的源代码

[[File:Dawson Integral.png|thumb|Dawson integral]]
'''道森积分'''（Dawson integral)由下式定义

:<math>F(x)  = e^{-x^2} \int_0^x e^{t^2}\,dt</math>,


==拐点==
道森积分的拐点为

-.92413887300459176701

+.92413887300459176701

==对称性==
[[File:Dawson Integral 2d.png|thumb|Anti symmetric  Dawson Integral]]
道森积分是反对称函数

<math>F(-x)=-F(x)</math>

==微商==
[[File:Dawson complex.gif|thumb|Derivative of Dawson Integral]]
道森积分的微商是
:<math>\frac{dF(x)}{dx}=1-2*x*F(x)</math>

==微分方程==
:<math>F'(y)=\int_0^{\infty}e^{-x^2}2x{\cos}(2xy)dx</math>
:<math>F'(y)+2yF(y)=\int_0^{\infty}2e^{-x^2}[x{\cos}(2xy)+y{\sin}(2xy)]dx=-e^{-x^2}{\cos}(2xy)|_0^{\infty}=1</math>
而且显然<math>F(0)=0</math>, 
因此<math>F(y)</math>是微分方程
:<math>f'(y)+2yf(y)=1</math>
在初始条件<math>f(0)=0</math>下的解，根据[[柯西-利普希茨定理]]解是唯一的.

==其他表达式==
:<math>F\left( y \right) = \int_0^y {{e^{{t^2} - {y^2}}}{\text{d}}t}</math>
证明:
只要证明也满足它的微分方程即可

对<math>y</math>求导，根据[[积分符号内取微分]]有
:<math>F'(y)=\frac{d}{dy}e^{-y^2}\int_0^{y}e^{t^2}dt=-2ye^{-y^2}\int_0^y e^{t^2}dt+1</math>


==参考文献==
*{{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.9. Dawson's Integral | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=302 | accessdate=2015-01-27 | archive-date=2011-08-11 | archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=302 }}
*{{dlmf|id=7|title=Error Functions, Dawson’s and Fresnel Integrals|first=N. M. |last=Temme}}

[[Category:特殊函数]]
[[Category:高斯函數]]