Inverse Erfc

Author

Eric W. Weisstein
September 20, 2008

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

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

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

Notation

v5.2

InverseErfc[x]

TraditionalForm[InverseErfc[x]]

v6

InverseErfc[x]

TraditionalForm[InverseErfc[x]]

Plot

<<Utilities`Plot`

Plot[InverseErfc[x],{x,0,2},PlotStyle->Red,AxesLabel->{x,InverseErfc[x]},PlotPoints->200]

Special Values

v5.2

InverseErfc/@{0,1,2}

N[%]

v6

InverseErfc/@{0,1,2}

Derivative

Plot

Plot[{-(1/2) (E^((InverseErf[∞,-z])^2)) (Sqrt[π]),(1/2) (E^(2 ((InverseErf[∞,-z])^2))) π InverseErf[∞,-z]},{z,0,2},PlotStyle->{Red,Blue},
PlotRange->{-5,5}]

v5.2

FullSimplify[D[InverseErfc[z],z]]

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

v6

FullSimplify[D[InverseErfc[z],z]]

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

Indefinite Integral

v5.2

Integrate[InverseErfc[z],z]

v6

Integrate[InverseErfc[z],z]

Identities

erfc((erfc^(-1))(x))==x

Erfc[InverseErfc[x]]

(erfc^(-1))(erfc(x))==x/;x∈ℝ

InverseErfc[Erfc[x]]

FullSimplify[InverseErfc[Erfc[x]],x∈Reals]

(erfc^(-1))(1-x)==(erf^(-1))(x)

FullSimplify[InverseErfc[1-x]]

FullSimplify[InverseErfc[1-x],-1<x<1]

FullSimplify[InverseErfc[1-x]-InverseErf[x],-1<x<1]

InverseErfc[1-x]-InverseErf[x]/.x->.02

Plot[{InverseErf[x],InverseErfc[1-x]},{x,-1,1},PlotStyle->{{Thickness[.02],Red},Blue}]

Series

About 0

v5.2

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

Series[InverseErfc[x],{x,1,5}]

Series[InverseErfc[x],{x,2,5}]

v6

Series[InverseErfc[x],{x,0,15}]

Series[InverseErfc[x],{x,0,20},Assumptions->x>0]

Series[InverseErfc[x],{x,1,10}]

Series[InverseErfc[x],{x,2,2}]

V7

Series[InverseErfc[x],{x,0,15},Assumptions->x∈Reals]