Dirichlet Eta Function

Author

Eric W. Weisstein
January 8, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Definition

Collect[Expand[Sum[((((-1))^(k-1))/(k^s)),{k,∞}]],Zeta[s]]

η[s_?NumericQ]:=(1-(2^(1-s))) Zeta[s]

Special Cases

η(0)

DirichletEta[0]

η(1)

DirichletEta[1]

Table

TextGrid[Table[{n,DirichletEta[n]},{n,5}],Dividers->All]//TraditionalForm

Derivative

Derivative[1][DirichletEta][x]

Plot[Derivative[1][DirichletEta][x],{x,-10,5},WorkingPrecision->20,PlotRange->2,PlotStyle->Red,AxesLabel->{x,η'[x]},BaseStyle->Directive[Black,FontFamily->"Times",12],LabelStyle->Directive[Black,14]]

η'(-1)==3 log(A)-(1/4)-((log(2))/3)

A271533

Derivative[1][DirichletEta][-1]//FullSimplify

N[%,10]

RealDigits[DirichletEta'[-1],10,110][[1]]

η'(0)==(1/2) log((π/2))

A256358

Derivative[1][DirichletEta][0]//FullSimplify

N[%,10]

η'((1/2))

A265162

Collect[Derivative[1][DirichletEta][(1/2)]//FunctionExpand,{Zeta[1/2]}]

((3-Sqrt[2])Log[2]/2-(Sqrt[2]-1)(2*EulerGamma+Pi+2Log[Pi])/4)Zeta[1/2]//TraditionalForm

N[%,10]

RealDigits[DirichletEta'[1/2],10,110][[1]]

η'(1) == γ log(2)-(((log^2)(2))/2)

A091812

Derivative[1][DirichletEta][1]

Limit[Derivative[1][DirichletEta][x],x->1]//FullSimplify

N[%,10]

RealDigits[Limit[Derivative[1][DirichletEta][x],x->1],10,110][[1]]

η'(2) == γ log(2)-(((log^2)(2))/2)

Derivative[1][DirichletEta][2]

N[%,10]

RealDigits[DirichletEta'[2],10,110][[1]]

Integral

Integrate[DirichletEta[x],x]

Plots

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

Real Axis

Plot[DirichletEta[x],{x,-10,10},
AxesLabel->TraditionalForm/@{x,DirichletEta[x]},
PlotStyle->Red]

Complex Plane

ComplexPlot3D[DirichletEta[z],{z,-15,15},PlotPoints->200]//Timing

ComplexContourPlot[DirichletEta[z],{z,-5,5}]//Timing

ComplexContourPlot[DirichletEta[z],{z,-15,15}]//Timing