Inverse Erf

Author

Eric W. Weisstein
September 20, 2008

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

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

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

Notation

v5.2

InverseErf[x]

TraditionalForm[InverseErf[x]]

v6

InverseErf[x]

TraditionalForm[InverseErf[x]]

V7

InverseErf[x]

TraditionalForm[InverseErf[x]]

Plot

<<Utilities`Plot`

Plot[InverseErf[x],{x,-1,1},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,HoldForm[InverseErf[x]]},PlotPoints->200]

Special Values

InverseErf/@{-1,0,1}

N[%]

Derivative and Integral

Derivative

Plot

Plot[{(1/2) (E^((InverseErf[z])^2)) (Sqrt[π]),(1/2) (E^(2 ((InverseErf[z])^2))) π InverseErf[z]},{z,-1,1},PlotStyle->{Red,Blue},PlotRange->{-10,10}]

v5.2

D[InverseErf[z],z]//FullSimplify

D[InverseErf[z],{z,2}]//FullSimplify

v6

D[InverseErf[z],z]//FullSimplify

D[InverseErf[z],{z,2}]//FullSimplify

v7

((∂(erf^(-1))(z))/(∂z))==(1/2) (Sqrt[π]) (E^(((erf^(-1))(z))^2))

D[InverseErf[z],z]

D[InverseErf[z],{z,2}]

Indefinite Integral

∫(erf^(-1))(z)dz==-((e^(-(((erf^(-1))(z))^2)))/(Sqrt[π]))

v5.2

Integrate[InverseErf[z],z]

v6

Integrate[InverseErf[z],z]

V7

Integrate[InverseErf[z],z]

Definite Integrals

(∫_0^1)(erf^(-1))(z)dz==(1/(Sqrt[π]))

v5.2

Integrate[InverseErf[z],{z,0,1}]

v6

Integrate[InverseErf[z],{z,0,1}]//Timing

V7

Integrate[InverseErf[z],{z,0,1}]

(∫_0^1)log((erf^(-1))(z))dz

InverseErf[z]

Plot[Log[InverseErf[z]],{z,0,1},PlotStyle->Red]

V5.2

Integrate[Log[InverseErf[z]],{z,0,1}]

V6

Integrate[Log[InverseErf[z]],{z,0,1}]//Timing

V7

Integrate[Log[InverseErf[z]],{z,0,1}]//Timing

Analytic (∫_t^(1-t))log((erf^(-1))(z))dz

V6

(expr=Integrate[Log[InverseErf[z]],{z,t,1-t},Assumptions->0<=t<1/2])//Timing

expr/.t->10^-10`20

Limit[expr,t->0]

V7

(expr=Integrate[Log[InverseErf[z]],{z,t,1-t},Assumptions->0<=t<1/2])//Timing

Limit[expr,t->0]

expr/.t->10^-10.

-(EulerGamma/2+Log[2])//N

Analytic (∫_((1/2)-t)^(t+(1/2)))log((erf^(-1))(z))dz

V5

(exprs=Integrate[Log[InverseErf[z]],{z,1/2-t,1/2+t},Assumptions->0<t<=1/2])//Timing

exprs/.t->1/2-10^-10`30

Limit[exprs,t->1/2]

exprs2=(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-1] 0)-(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-((InverseErf[0,(1/2)+t])^2)] InverseErf[0,(1/2)+t])-Erf[1] Log[1]+Erf[InverseErf[0,(1/2)+t]] Log[InverseErf[0,(1/2)+t]];

exprs2/.t->1/2-10^-10`30

FullSimplify[exprs2,0<t≤1/2]

-(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-((InverseErf[0,(1/2)+t])^2)] InverseErf[0,(1/2)+t])+(t+1/2) Log[InverseErf[0,(1/2)+t]]/.t->1/2-10^-10`30

FullSimplify[-(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-((InverseErf[0,(1/2)+t])^2)] InverseErf[0,(1/2)+t])+(t+1/2) Log[InverseErf[0,(1/2)+t]]]

expr3=-(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-((InverseErf[0,(1/2)+t])^2)] InverseErf[0,(1/2)+t])+((1/2)+t) Log[InverseErf[0,(1/2)+t]];

expr3/.t->1/2-10^-10`20

expr4=expr3/.t->x-1/2

expr4/.x->1-10^-10`20

Limit[-(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-((InverseErf[0,x])^2)] InverseErf[0,x])+x Log[InverseErf[0,x]],x->1]

Limit[-(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-((InverseErf[0,x])^2)] InverseErf[0,x])+x Log[InverseErf[0,x]],x->1]

ser=Series[-(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-((InverseErf[0,x])^2)] InverseErf[0,x])+x Log[InverseErf[0,x]],{x,1,0}]

Solve[y==InverseErf[0,x],x]

-(1/(Sqrt[π]))(2 HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-((InverseErf[0,x])^2)] InverseErf[0,x])+x Log[InverseErf[0,x]]/.{InverseErf[0,x]->y,x->Erf[y]}

N[-((2 y HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-(y^2)])/(Sqrt[π]))+Erf[y] Log[y]/.y->10,10]

Limit[-((2 y HypergeometricPFQ[{(1/2),(1/2)},{(3/2),(3/2)},-(y^2)])/(Sqrt[π]))+Erf[y] Log[y],y->Infinity]

N[1/2PolyGamma[0,1/2]]

FunctionExpand[PolyGamma[0,1/2]/2]//FullSimplify

V7

(exprs=Integrate[Log[InverseErf[z]],{z,1/2-t,1/2+t},Assumptions->0<t<=1/2])//Timing

Limit[exprs,t->1/2]

Numerical v5.2

NIntegrate[Log[InverseErf[z]],{z,0,1}]

NIntegrate[Log[InverseErf[z]],{z,0,1},WorkingPrecision->10]//Timing

NIntegrate[Log[InverseErf[z]],{z,0,1},Method->DoubleExponential,WorkingPrecision->10]//Timing

NIntegrate[Log[InverseErf[z]],{z,0,1},WorkingPrecision->20]//Timing

NIntegrate[Log[InverseErf[z]],{z,0,1},WorkingPrecision->50]//Timing

Options[NIntegrate]

NIntegrate[Log[InverseErf[z]],{z,0,1},WorkingPrecision->50,SingularityDepth->7,MaxRecursion->10]//Timing

NIntegrate[Log[InverseErf[z]],{z,0,1},Method->MonteCarlo,PrecisionGoal->5,MaxPoints->10^6]//Timing

NIntegrate[Log[InverseErf[z]],{z,0,1},Method->QuasiMonteCarlo,PrecisionGoal->5,MaxPoints->10^6]//Timing

Series

Series[Log[InverseErf[z]],{z,0,10}]

<<DiscreteMath`RSolve`

SeriesTerm[Log[InverseErf[x]],{x,0,n}]

CoefficientList[Series[Log[InverseErf[z]],{z,0,20}],z^2]

{n,d}=Through[{Numerator,Denominator}[%]]

Rest[n]/.Pi->1

Identities

Erf[InverseErf[x]]

InverseErf[Erf[x]]

FullSimplify[InverseErf[Erf[x]],x∈Reals]

Series

(erf^(-1))(((2 (x-1))/(Sqrt[π])))

s2=Series[InverseErf[((2(x-1))/(Sqrt[π]))],{x,1,24}]

(erf^(-1))(((2 x)/(Sqrt[π])))

s1=Series[InverseErf[((2x)/(Sqrt[π]))],{x,0,30}]

Recurrence

c[0]:=1
c[k_]:=c[k]=Sum[((c[m]c[k-1-m])/((m+1)(2m+1))),{m,0,k-1}]

Table[((c[n])/(2n+1)),{n,0,10}]

Terms

s1=Series[InverseErf[((2x)/(Sqrt[π]))],{x,0,30}]

l1=Table[(2n+1)!/2^n,{n,0,14}]

A002067 ={1, 1, 7, 127, 4369, 243649, 20036983, 2280356863, 343141433761, 65967241200001, 15773461423793767, 4591227123230945407, 1598351733247609852849, 655782249799531714375489, 313160404864973852338669783};

A002067 /l1

v5.2

<<DiscreteMath`RSolve`

SeriesTerm[InverseErf[x],{x,0,n}]

Series[InverseErf[x],{x,0,10}]

SeriesTerm[InverseErf[((2x)/(Sqrt[π]))],{x,0,n}]

Last/@Partition[CoefficientList[s1,x],2]

Last/@Partition[CoefficientList[s2/.x->z+1,z],2]

Through[{Numerator,Denominator}[s1]]

Series[Erf[x],{x,0,5}]

InverseSeries[Series[Erf[x],{x,0,5}]]

v6

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

SeriesCoefficient[InverseErf[((2x)/(Sqrt[π]))],{x,0,n}]

v7

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

SeriesCoefficient[InverseErf[((2x)/(Sqrt[π]))],{x,0,n}]

Series[InverseErf[((2x)/(Sqrt[π]))],{x,0,30}]

Through[{Numerator,Denominator}[CoefficientList[%,x][[2;;-1;;2]]]]