道森積分

道森積分（Dawson integral)由下式定義

${\displaystyle F(x)=e^{-x^{2}}\int _{0}^{x}e^{t^{2}}\,dt}$,

Dawson integral

拐點

道森積分的拐點為

-.92413887300459176701

+.92413887300459176701

對稱性

Anti symmetric Dawson Integral

道森積分是反對稱函數

${\displaystyle F(-x)=-F(x)}$

微商

Derivative of Dawson Integral

道森積分的微商是

${\displaystyle {\frac {dF(x)}{dx}}=1-2*x*F(x)}$

微分方程

${\displaystyle F'(y)=\int _{0}^{\infty }e^{-x^{2}}2x{\cos }(2xy)dx}$

${\displaystyle F'(y)+2yF(y)=\int _{0}^{\infty }2e^{-x^{2}}[x{\cos }(2xy)+y{\sin }(2xy)]dx=-e^{-x^{2}}{\cos }(2xy)|_{0}^{\infty }=1}$

而且顯然${\displaystyle F(0)=0}$, 因此${\displaystyle F(y)}$是微分方程

${\displaystyle f'(y)+2yf(y)=1}$

在初始條件${\displaystyle f(0)=0}$下的解，根據柯西-利普希茨定理解是唯一的.

其他表達式

${\displaystyle F\left(y\right)=\int _{0}^{y}{{e^{{t^{2}}-{y^{2}}}}{\text{d}}t}}$

證明: 只要證明也滿足它的微分方程即可

對${\displaystyle y}$求導，根據積分符號內取微分有

${\displaystyle F'(y)={\frac {d}{dy}}e^{-y^{2}}\int _{0}^{y}e^{t^{2}}dt=-2ye^{-y^{2}}\int _{0}^{y}e^{t^{2}}dt+1}$

參考文獻

• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP, Section 6.9. Dawson's Integral, Numerical Recipes: The Art of Scientific Computing 3rd, New York: Cambridge University Press, 2007 [2015-01-27], ISBN 978-0-521-88068-8, （原始內容存檔於2011-08-11）
• Temme, N. M., Error Functions, Dawson’s and Fresnel Integrals, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248