Dirichlet Beta Function

Author

Eric W. Weisstein
February 17, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Definition

DirichletBeta[x]

V6

Sum[(-1)^n(((2n+1))^(-x)),{n,0,∞}]

Limit[(2^(-2 x)) (Zeta[x,(1/4)]-Zeta[x,(3/4)]),x->1]//FullSimplify

Sum[(-1)^n(2n+1)^(-1),{n,0,∞}]

V7

Sum[(-1)^n(((2n+1))^(-x)),{n,0,∞}]

Limit[(2^(-2 x)) (Zeta[x,(1/4)]-Zeta[x,(3/4)]),x->1]//FullSimplify

Sum[(-1)^n(2n+1)^(-1),{n,0,∞}]

V10

bug(265447)

sum=Sum[(-1)^n(((2n+1))^(-x)),{n,0,∞}]

FullSimplify[sum]

FullSimplify[%-DirichletBeta[x]]

FunctionExpand[DirichletBeta[x]]

Plots

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

Real Axis

Plot[DirichletBeta[x],{x,-6,6},AxesLabel->TraditionalForm/@{x,DirichletBeta[x]},
PlotStyle->Red]

Complex Plane

ComplexPlot3D[DirichletBeta[z],{z,-2π,2π},PlotPoints->200]//Timing

ComplexContourPlot[DirichletBeta[z],{z,-2π,2π}]//Timing

Special Values

Negative Odd n

FullSimplify[DirichletBeta[Range[-1,-20,-2]]]

DirichletBeta[Range[-1,-20,-2]]

Assuming[n∈Integers&&n<0&&Mod[n,2]==1,FullSimplify[FunctionExpand[DirichletBeta[n]]]]

Negative Even n

Table[((EulerE[-k])/2),{k,-2,-20,-2}]

DirichletBeta[Range[-2,-20,-2]]

Assuming[n∈Integers&&n<0&&Mod[n,2]==0,FullSimplify[FunctionExpand[DirichletBeta[n]]]]

Odd positive n

Assuming[n∈Integers&&n>0&&Mod[n,2]==1,FullSimplify[FunctionExpand[DirichletBeta[n]]]]

Table[(((((-1))^k)EulerE[2k])/(2(2k)!))((((π/2)))^(2k+1)),{k,1,10,2}]

FullSimplify[DirichletBeta[Range[1,20,2]]]

FullSimplify[%]

Through[{Numerator,Denominator}[%]]

β(1)==(π/4)

Numeric

DirichletBeta[x]/.x->1.00001

Pi/4.

V6

Limit[DirichletBeta[x],x->1]

FullSimplify[%]

V10

DirichletBeta[1]

β(∞)==1

Numeric

DirichletBeta[x]/.x->999.

v4.2

Limit[DirichletBeta[x],x->Infinity]

v5.0

Limit[DirichletBeta[x],x->Infinity]

v5.2

Limit[DirichletBeta[x],x->Infinity]

v6

Limit[DirichletBeta[x],x->Infinity]

V10

bug(265449)

DirichletBeta[∞]

Limit[DirichletBeta[x],x->∞]

DirichletBeta[100.]

β(n)

Old

Beta2[x_]=2^(-x)LerchPhi[-1,x,1/2]

FunctionExpand[FullSimplify[Table[Beta2[n]/.Zeta[x_,nn_,o___]:>Zeta[x,nn],{n,5}]]]

FullSimplify[(1/768) (-8 (π^4)+PolyGamma[3,(1/4)])-((PolyGamma[3,(1/4)]-PolyGamma[3,(3/4)])/1536)]

FullSimplify[FunctionExpand[Table[DirichletBeta[n],{n,1,10}]]]

v2=Table[(-1)^k EulerE[2k](Pi/2)^(2k+1)/2/
(2k)!,{k,0,5}]

V10

Table[DirichletBeta[n],{n,4}]//FullSimplify

Sums

<<Utilities`Simplify`

Sum[Log[((((4k+1))^(1/(((4k+1))^n)))/(((4k-1))^(1/(((4k-1))^n))))],{k,∞}]

FullSimplify[-D[(2^(-2 x)) (Zeta[x,(1/4)]-Zeta[x,(3/4)]),x]/.x->n,n∈Integers&&n>0]

(∑_(k=1)^∞)log(((((4 k+1))^(1/(4 k+1)))/(((4 k-1))^(1/(4 k-1)))))==-β'(1)

V6

Sum[Log[((((4k+1))^(1/(4k+1)))/(((4k-1))^(1/(4k-1))))],{k,∞}]//Expand

Total[N[Table[Log[((((4k+1))^(1/(4k+1)))/(((4k-1))^(1/(4k-1))))],{k,10^4}]]]

(1/4) π (log(((2 π ((Gamma[(3/4)])^2))/((Gamma[(1/4)])^2)))+ℽ)//N

V7

Sum[Log[((((4k+1))^(1/(4k+1)))/(((4k-1))^(1/(4k-1))))],{k,∞}]//GammaSimplify//LogExpand

(∑_(k=1)^∞)log(((((4 k+1))^(1/(((4 k+1))^2)))/(((4 k-1))^(1/(((4 k-1))^2)))))==-β'(2)

Sum[Log[((((4k+1))^(1/(((4k+1))^2)))/(((4k-1))^(1/(((4k-1))^2))))],{k,∞}]//Expand

Sum[Log[((((4k+1))^(1/(((4k+1))^2)))/(((4k-1))^(1/(((4k-1))^2))))],{k,∞}]//Expand

-Derivative[1][DirichletBeta][2]//FullSimplify//LogExpand//Expand

(∑_(k=1)^∞)log(((((4 k+1))^(1/(((4 k+1))^3)))/(((4 k-1))^(1/(((4 k-1))^3)))))==-β'(3)

Sum[Log[((((4k+1))^(1/(((4k+1))^3)))/(((4k-1))^(1/(((4k-1))^3))))],{k,∞}]

-Derivative[1][DirichletBeta][3]//FullSimplify//LogExpand

(∑_(k=1)^∞)log(((((4 k+1))^(1/(((4 k+1))^4)))/(((4 k-1))^(1/(((4 k-1))^4)))))==-β'(4)

Sum[Log[((((4k+1))^(1/(((4k+1))^4)))/(((4k-1))^(1/(((4k-1))^4))))],{k,∞}]

-Derivative[1][DirichletBeta][3]//FullSimplify//LogExpand

Derivatives

Directly

FunctionExpand[Derivative[1][DirichletBeta][x]]

Via Sum

D[(((-1))^n)(((2n+1))^(-x)),x]

Sum[-(((-1))^n) (((1+2 n))^(-x)) Log[1+2 n],{n,0,∞}]

β'(-1)

Old

Derivative[1][DirichletBeta][-1]

FullSimplify[%]

N[%,10]

V10

Derivative[1][DirichletBeta][-1]

FullSimplify[%]

β'(0)

Old

Derivative[1][DirichletBeta][0]

%//FullSimplify

<<Utilities`Simplify`

GammaSimplify[Log[((2 Gamma[(5/4)])/(Gamma[(3/4)]))]]

N[%,20]

ListPlot[Table[-(((-1))^n) Log[1+2 n],{n,0,50}],PlotStyle->{Red,Thickness[.02]}]

V6

Sum[-(((-1))^n) Log[1+2 n],{n,0,∞}]

V7

-(((-1))^n) (((1+2 n))^(-x)) Log[1+2 n]/.x->0

Sum[-(((-1))^n) Log[1+2 n],{n,0,∞}]

NSum[-(((-1))^n) Log[1+2 n],{n,0,∞}]

Sum[-(((-1))^n) Log[1+2 n],{n,0,10^4}]//N

V10

Derivative[1][DirichletBeta][0]

FullSimplify[%]

β'(1)==(1/4) π (log(((2 π ((Γ((3/4)))^2))/((Γ((1/4)))^2)))+ℽ)

Old

Derivative[1][DirichletBeta][1]

Derivative[1][DirichletBeta][1.+10^-10]

N[(1/4) π (EulerGamma+Log[((2 π ((Gamma[(3/4)])^2))/((Gamma[(1/4)])^2))])]

v5.2

l=Limit[Derivative[1][DirichletBeta][x],x->1]

v6

Limit[Derivative[1][DirichletBeta][x],x->1]

V10

bug(265453)

Derivative[1][DirichletBeta][1]

Limit[Derivative[1][DirichletBeta][x],x->1]

FullSimplify[%]

FunctionExpand[%]

N[%]

(1/4)Pi(EulerGamma+2Log[2]+3Log[π]-4Log[Gamma[1/4]])//N

β'(1-2n)

Benoit Cloitre <abmt@wanadoo.fr>
Wed, 25 Jan 2006 15:03:03 -0600

Derivative[1][DirichletBeta][1-2n]//FullSimplify

Table[Derivative[1][DirichletBeta][1-2n],{n,1}]//FullSimplify

Table[(n+(1/2))EulerE[2n]((DirichletBeta[2n])/(DirichletBeta[2n-1])),{n,1}]//FullSimplify

...not quite right...

N[%]

β''(1)

v5.2

Limit[Derivative[1][DirichletBeta][x],x->1]//InputForm

l=Limit[Derivative[2][DirichletBeta][x],x->1]

v6

l=Limit[Derivative[2][DirichletBeta][x],x->1]

FullSimplify[FunctionExpand[l]]

GammaSimplify[FullSimplify[FunctionExpand[l]]]

V10

Derivative[2][DirichletBeta][1]

Limit[Derivative[2][DirichletBeta][x],x->1]

FullSimplify[%]

FunctionExpand[%]

N[%]