Riemann Zeta Function ζ(2)

Author

Eric W. Weisstein
September 28, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/RiemannZetaFunctionZeta2.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Value

Zeta[2]

N[Zeta[2]]

Identities

Sum[(1/(k^2)),{k,n}]

FunctionExpand[%]

Series

(∑_(k=1)^∞)(1/(k^2))

Sum[1/k^2, {k, Infinity}]

3 (∑_(k=1)^∞)(1/((k^2) (({{2 k}, {k}}))))

V6

3Sum[(1/((k^2)Binomial[2k,k])),{k,∞}]

Double Series

(∑_(i=1)^∞)(∑_(j=1)^∞)(((i-1)! (j-1)!)/((i+j)!))

"CLOITRE Benoit" <abcloitre@wanadoo.fr>
Date: Thu, 9 Dec 2004 14:18:21 +0100

V6

Sum[(((i-1)!(j-1)!)/((i+j)!)),{i,∞},{j,∞}]

Sum[(((i-1)!(j-1)!)/((i+j)!)),{i,∞}]

FullSimplify[%]

NSum[(((i-1)!(j-1)!)/((i+j)!)),{j,∞},{i,∞}]//Timing

NSum[(((i-1)!(j-1)!)/((i+j)!)),{j,∞},{i,∞},Method->Integrate,VerifyConvergence->False]//Timing

V7

TimeConstrained[Sum[(((i-1)!(j-1)!)/((i+j)!)),{i,∞},{j,∞}],60]//Timing

TimeConstrained[Sum[(((i-1)!(j-1)!)/((i+j)!)),{j,∞},{i,∞}],60]//Timing

Sum[(((i-1)!(j-1)!)/((i+j)!)),{i,∞}]

FullSimplify[%]

TimeConstrained[NSum[(((i-1)!(j-1)!)/((i+j)!)),{j,∞},{i,∞}],60]//Timing

Fourier Series

V6

FourierA[0,(x^(2n)),{x,-π,π},Assumptions->n∈Integers&&n>=0]

FourierA[m,(x^(2n)),{x,-π,π},Assumptions->(m|n)∈Integers&&n≥0]

Simplify[Table[(1/(1+2 n))(2 (π^(2 n)) HypergeometricPFQ[{(1/2)+n},{(1/2),(3/2)+n},-(1/4) (m^2) (π^2)]),{n,3}],m∈Integers&&m>0]

FourierB[m,(x^(2n)),{x,-π,π},Assumptions->(m|n)∈Integers&&n>=0]

((π^(2 n))/(1+2 n))+(∑_m^∞)((2 (π^(1+2 n)) Cos[m x] HypergeometricPFQ[{(1/2)+n},{(1/2),(3/2)+n},-(1/4) (m^2) (π^2)])/(1+2 n))

TimeConstrained[With[{n=1},(1/2)((2 (π^(2 n)))/(1+2 n))+Sum[((2 (π^(2 n)) HypergeometricPFQ[{(1/2)+n},{(1/2),(3/2)+n},-(1/4) (m^2) (π^2)])/(1+2 n))Cos[m x],{m,∞}]],60]//FullSimplify//Timing

TimeConstrained[With[{n=2},(1/2)((2 (π^(2 n)))/(1+2 n))+Sum[((2 (π^(2 n)) HypergeometricPFQ[{(1/2)+n},{(1/2),(3/2)+n},-(1/4) (m^2) (π^2)])/(1+2 n))Cos[m x],{m,∞}]],60]//FullSimplify//Timing

V7

FourierA[0,(x^(2n)),{x,-π,π},Assumptions->n∈Integers&&n>=0]

FourierA[m,(x^(2n)),{x,-π,π},Assumptions->(m|n)∈Integers&&n≥0]

Simplify[Table[(1/(1+2 n))(2 (π^(2 n)) HypergeometricPFQ[{(1/2)+n},{(1/2),(3/2)+n},-(1/4) (m^2) (π^2)]),{n,3}],m∈Integers&&m>0]

FourierB[m,(x^(2n)),{x,-π,π},Assumptions->(m|n)∈Integers&&n>=0]

(1/2)((2 (π^(2 n)))/(1+2 n))+Sum[((2 (π^(2 n)) HypergeometricPFQ[{(1/2)+n},{(1/2),(3/2)+n},-(1/4) (m^2) (π^2)])/(1+2 n))Cos[m x],{m,∞}]

TimeConstrained[With[{n=1},(1/2)((2 (π^(2 n)))/(1+2 n))+Sum[((2 (π^(2 n)) HypergeometricPFQ[{(1/2)+n},{(1/2),(3/2)+n},-(1/4) (m^2) (π^2)])/(1+2 n))Cos[m x],{m,∞}]],60]//Timing

TimeConstrained[With[{n=2},(1/2)((2 (π^(2 n)))/(1+2 n))+Sum[((2 (π^(2 n)) HypergeometricPFQ[{(1/2)+n},{(1/2),(3/2)+n},-(1/4) (m^2) (π^2)])/(1+2 n))Cos[m x],{m,∞}]],60]//Timing

FullSimplify[(1/3) ((π^2)+6 PolyLog[2,-(E^(-I x))]+6 PolyLog[2,-(E^(I x))]),-π<x<π]

FullSimplify[(1/3) ((π^2)+6 PolyLog[2,-(E^(-I x))]+6 PolyLog[2,-(E^(I x))])-x^2,-π<x<π]

Table[(1/3) ((π^2)+6 PolyLog[2,-(E^(-I x))]+6 PolyLog[2,-(E^(I x))])-x^2/.x->RandomReal[π{-1,1}],{100}]//Chop//Union

Plot[(1/3) ((π^2)+6 PolyLog[2,-(E^(-I x))]+6 PolyLog[2,-(E^(I x))]),{x,-5,5}]

Integrals

(∫_0^1)(∫_0^1)(1/(1-x y))dydx

Integrate[(1/(1-x y)),{x,0,1},{y,0,1}]

(∫_0^1)(∫_0^1)(1/(1-x y))dxdy

Integrate[(1/(1-x y)),{y,0,1},{x,0,1}]

(Subscript[I, 1])

i1=4Integrate[(1/(2-(u^2)+(v^2))),{u,0,(Sqrt[2])/2},{v,0,u}]

4Integrate[(1/(Sqrt[2-(u^2)]))ArcTan[(u/(Sqrt[2-(u^2)]))],{u,0,(Sqrt[2])/2}]

(Subscript[I, 2])

i2=4Integrate[(1/(2-(u^2)+(v^2))),{u,(Sqrt[2])/2,(Sqrt[2])},{v,0,(Sqrt[2])-u}]

4Integrate[(1/(Sqrt[2-(u^2)]))ArcTan[(((Sqrt[2])-u)/(Sqrt[2-(u^2)]))],{u,(Sqrt[2])/2,(Sqrt[2])}]

FullSimplify[ArcTan[(((Sqrt[2])-u)/(Sqrt[2-(u^2)]))]]

Plot[ArcTan[(((Sqrt[2])-u)/(Sqrt[2-(u^2)]))]-(1/2)((1/2)π-ArcSin[u/(Sqrt[2])]),{u,-1.5,1.5}]

4Integrate[(1/((Sqrt[2])Cos[θ]))((1/4)θ-(1/2)θ)(Sqrt[2])Cos[θ],{θ,π/6,π/2}]

i1+i2