Dawson's Integral

Author

Eric W. Weisstein
August 26, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/DawsonsIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DawsonsIntegral.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Analytic Values

{Exp[-x^2]Integrate[Exp[y^2],{y,0,x}],Exp[x^2]Integrate[Exp[-y^2],{y,0,x}]}

FunctionExpand[DawsonF[x]]

FullSimplify[%]

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

V6

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
{
Plot[DawsonF[1,x],{x,-5,5},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,DawsonF[1,x]}],Plot[DawsonF[-1,x],{x,-1.5,1.5},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,DawsonF[-1,x]}]}]
]
]

V7

GraphicsGrid[{{Plot[DawsonF[x],{x,-5,5},PlotStyle->Red,AxesLabel->{x,DawsonF[x]}],
Plot[DawsonIntegral[-1,x],{x,-1.5,1.5},
PlotStyle->Red,AxesLabel->{x,Subscript[D,"-"][x]}]}},ImageSize->400]

(SubPlus[D])

ComplexPlot3D[DawsonF[1,z],{z,-2,2},PlotPoints->100]//Timing

ComplexContourPlot[DawsonF[1,z],{z,-2,2}]//Timing

(SubMinus[D])

ComplexPlot3D[DawsonF[-1,z],{z,-2,2},PlotPoints->100]//Timing

ComplexContourPlot[DawsonF[-1,z],{z,-2,2}]//Timing

Identities

(DawsonF[-x])==-(DawsonF[x])

DawsonF[-x]

DawsonF[-x]+DawsonF[x]

(DawsonF[x])==(1/2) i (Sqrt[π]) (e^(-(x^2))) erf(i x)

FullSimplify[DawsonF[x]+I Sqrt[π]/2Exp[-x^2]Erf[I x]]

Differential Equation

F'(z)+2 z F(z)==1

DSolve[F'[z]+2z F[z]==1,F[z],z]

DSolve[F'[z]+2z F[z]==1,F[z],z]

FunctionExpand[DawsonF[x]]

Integrals

(∫_0^x)exp((t^2)-(x^2))dt==(DawsonF[x])

Integrate[Exp[t^2-x^2],{t,0,x}]

FunctionExpand[DawsonF[x]]

Continued Fractions

(z/(ContinuedFractionK[2 (((-1))^(n+1)) n (z^2),2 n+1,{n,∞}]+1))

z/(1+ContinuedFractionK[(-1)^(n+1) 2n z^2,2n+1,{n,∞}])

z/(1+ContinuedFractionK[(-1)^(n+1) 2n z^2,2n+1,{n,20}])/.z->1.1

DawsonF[1.1]

(z/(ContinuedFractionK[-4 n (z^2),(2 n+1)+2 (z^2),{n,∞}]+2 (z^2)+1))

z/(1+2z^2+ContinuedFractionK[- 4n z^2,(2n+1)+2z^2,{n,∞}])

z/(1+2z^2+ContinuedFractionK[- 4n z^2,(2n+1)+2z^2,{n,20}])/.z->1.1

Series

(lim_(x->0)) (DawsonF[x])

Series[DawsonF[x],{x,0,20}]

Through[{Numerator,Denominator}[CoefficientList[Series[DawsonF[x]/x,{x,0,30}],x^2]]]

SeriesCoefficient[DawsonF[x],{x,0,n}]

Table[(((-2)^n)/((2n+1)!!))(x^(2n+1)),{n,0,5}]

Table[Piecewise[{{((((-2)^((n-1)/2)))/(n!!))x^n,Mod[n,2]==1}}],{n,0,10}]

(lim_(x->∞)) (DawsonF[x])

Series[DawsonF[x],{x,Infinity,5}]

Expand[%]

Series[FunctionExpand[DawsonF[x]],{x,Infinity,5}]//Expand//Normal

{DawsonF[x],(1/(2x))+(1/(4(x^3))),(1/(2x))+(1/(4(x^2)))}/.x->100.

DawsonF[100.]

Series[DawsonF[x],{x,Infinity,2}]

%/.x->100

Series[DawsonF[x],{x,Infinity,5}]/.x->100

(Series[DawsonF[x],{x,Infinity,5}]//Normal)

N[%/.x->100]

(SubMinus[D])

Limit[Dawson[-1,x],x->∞]

Series[Dawson[-1,x],{x,0,3}]

Sums

(1/2) (Sqrt[π x]) (∑_(k=0)^∞)((((-x))^k)/(Γ(k+(3/2))))==(DawsonF[(Sqrt[x])])

((Sqrt[π x])/2)Sum[((((-x))^k)/(Gamma[k+(3/2)])),{k,0,∞}]

((x(Sqrt[π ]))/2)Sum[(((((-1))^k)(x^(2k)))/(Gamma[k+(3/2)])),{k,0,∞}]

(Sqrt[x]) (∑_(k=0)^∞)((((-x))^k)/(Pochhammer[(3/2), k]))==(DawsonF[(Sqrt[x])])

Sqrt[x]Sum[((((-x))^k)/(Pochhammer[3/2,k])),{k,0,∞}]

x Sum[(((((-1))^k)(x^(2k)))/(Pochhammer[3/2,k])),{k,0,∞}]

Derivative

D[DawsonF[x],x]

D[DawsonF[b x],x]

D[FunctionExpand[DawsonF[b x]],x]

FullSimplify[%-%%]

D[DawsonF[(Sqrt[b x])],x]

FullSimplify[%]

Integrate

Integrate[DawsonF[x],x]

(1/2)(x^2)Sum[((Pochhammer[1,k])/(Pochhammer[3/2,k]Pochhammer[2,k]))(((-(x^2)))^k),{k,0,∞}]

Relations

FunctionExpand[DawsonF[x]]

Seems to be wrong in Spanier and Oldham

FunctionExpand[x HypergeometricU[1,3/2,-x^2]]//FullSimplify

FullSimplify[FunctionExpand[x HypergeometricU[1,3/2,-x^2]],Assumptions->x<0]

Critical Points

d[i_]:=D[Dawson[i,x],x]
d2[i_]:=D[Dawson[i,x],{x,2}]

Maximum

Plot[DawsonF'[x],{x,-5,5},PlotRange->All]

FindInstance[DawsonF'[x]==0&&0<x<1,x,Reals]

FunctionExpand[DawsonF'[x]]

max = x/. FindRoot[1-(E^(-(x^2))) (Sqrt[π]) x Erfi[x],{x,.7},WorkingPrecision->30]

DawsonF[max]

Solve[1+(E^(y^2)) (Sqrt[π]) y Erf[y]==0,y]

Maximize[{DawsonF[x],x>0},x]

Maximize[{DawsonF[x],0<x<1},x]

NMaximize[{DawsonF[x],0<x<1},x]

Inflection Points

Plot[DawsonF''[x],{x,-5,5},PlotRange->All]

FindInstance[DawsonF''[x]==0&&0<x<2,x,Reals]//FullSimplify

infl=x/.FindRoot[d2[1],{x,1},WorkingPrecision->30]

FullSimplify[d2[1]]

Dawson[1,infl]

FullSimplify[D[DawsonF[1,x],{x,2}]]

Solve[%==0,x]

Reduce[-2 x+(E^(-(x^2))) (Sqrt[π]) (-1+2 (x^2)) Erfi[x]==0&&0<x<2,x,Reals]