Owen T-Function

Author

Eric W. Weisstein
December 1, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/OwenT-Function.nb.

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Plot

Plot[Evaluate@Table[OwenT[x,a],{a,0,5}],{x,-4,4},PlotRange->Full,PlotLegends->Table[T[x,n],{n,0,5}],BaseStyle->Directive[Black,FontFamily->"Times"],LabelStyle->Directive[Black,FontFamily->"Times"]]

Integral

Integrate[((Exp[-(x^2)(1+(t^2))/2])/(1+(t^2))),{t,0,a}]//Head//Timing

Assuming[a∈Reals,Integrate[((Exp[-(x^2)(1+(t^2))/2])/(1+(t^2))),{t,0,a}]]//Head//Timing

Integrate[((Exp[-(x^2)(1+(t^2))/2])/(1+(t^2))),{t,0,0}]

Integrate[((Exp[-(x^2)(1+(t^2))/2])/(1+(t^2))),{t,0,1}]//Head//Timing

Equal@@@N[Table[{
OwenT[x,1],
(1/(2π))NIntegrate[((Exp[-(x^2)(1+(t^2))/2])/(1+(t^2))),{t,0,1},WorkingPrecision->20],
(1/8) Erfc[-(x/(Sqrt[2]))] Erfc[(x/(Sqrt[2]))]
},{x,-2,2,1/10}]]//Union

Special Values

OwenT[x, n]

TextGrid[Table[{n,OwenT[x,n],FunctionExpand@OwenT[x,n]},{n,0,3}],Dividers->All]

Symmetry

OwenT[-x,a]

OwenT[x,-a]