Abel's Integral

Author

Eric W. Weisstein
January 25, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/AbelsIntegral.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Exact

N[1/2 Log[2]-1/4,20]

Forms

(∫_0^∞)(t/(((t^2)+1) ((e^(π t))-(e^(-π t)))))dt

Plot[t/(1+t^2)/(Exp[Pi t]-Exp[-Pi t]),{t,0,3},PlotStyle->Red,PlotRange->All]

NIntegrate[t/(1+t^2)/(Exp[Pi t]-Exp[-Pi t]),{t,0,Infinity}]

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

Integrate[(t/((E^(-π t))(-1+(E^(2π t))) (1+(t^2)))),{t,0,∞}]

(1/2)(∫_(-∞)^∞)(t/(((t^2)+1) ((e^(π t))-(e^(-π t)))))dt

(1/2)NIntegrate[(t/((E^(-π t))(-1+(E^(2π t))) (1+(t^2)))),{t,-∞,∞}]

(1/2) (∫_1^∞)(1/(u ((e^(π (Sqrt[u-1])))-(e^(-π (Sqrt[u-1]))))))du

(1/2)NIntegrate[(1/(u(Exp[π(Sqrt[u-1])]-Exp[-π(Sqrt[u-1])]))),{u,1,∞}]

Integrate[(1/(u(Exp[π(Sqrt[u-1])]-Exp[-π(Sqrt[u-1])]))),{u,1,∞}]

FullSimplify[(1/(u(Exp[π(Sqrt[u-1])]-Exp[-π(Sqrt[u-1])])))]

Integrate[((Csch[π (Sqrt[-1+u])])/(2 u)),{u,1,∞}]//Timing

(1/2) (∫_0^∞)((t csch(π t))/((t^2)+1))dt

ExpToTrig[t/(1+t^2)/(Exp[Pi t]-Exp[-Pi t])]

FullSimplify[t/(1+t^2)/(Exp[Pi t]-Exp[-Pi t])]

Plot[((t Csch[π t])/(2+2 (t^2))),{t,0,3},PlotStyle->Red,PlotRange->All]

(1/2)NIntegrate[((t Csch[π t])/(1+(t^2))),{t,0,Infinity}]

Integrate[((t Csch[π t])/(2+2 (t^2))),{t,0,∞}]//Timing

(1/(π^2))(∫_0^1)((log(x))/(((x^2)-1) ((((((log(x))/π)))^2)+1)))dx

Solve[x==Exp[Pi t],t]

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

Plot[(1/(π^2))((Log[x])/((x^2-1)((Log[x]/π)^2+1))),{x,0,1},PlotStyle->Red]

(1/(π^2))NIntegrate[((Log[x])/((x^2-1)((Log[x]/π)^2+1))),{x,0,1}]

Integrate[-(1/(π^2))(1/( ((1/(Log[x]))+((Log[x])/(π^2)))))(x^(2n)),{x,0,1},Assumptions->n≥0&&n∈Integers]

(∑_(n=1)^∞)Ci(π (2 n+1))

FullSimplify[Table[CosIntegral[π+2 n π],{n,3}]]

Sum[CosIntegral[π(2 n +1)],{n,0,200}]//N[#,20]&

(∫_0^∞)((csch((π/x)))/(2 x ((x^2)+1)))dx

FullSimplify[((t Csch[π t])/(2+2 (t^2)))/.t->1/x]

Plot[((Csch[(π/x)])/(2x(1+ (x^2)))),{x,0,10},PlotStyle->Red,PlotRange->All]

NIntegrate[(( Csch[(π/x)])/(2x(1+(x^2)))),{x,0,∞}]

Integrate[(( Csch[(π/x)])/(2x(1+(x^2)))),{x,0,∞}]//Timing

Integrate[((1/(2 x))-(x/(2 (1+(x^2)))))Csch[(π/x)],{x,0,∞}]

Series

<<DiscreteMath`RSolve`

Series[(1/(((x^2)-1))),{x,0,7}]

FullSimplify[-SeriesTerm[(1/((x^2)-1)),{x,0,2n}],n∈Integers&&n>0]

Series[((t )/(2+2 (t^2))),{t,0,5}]

SeriesTerm[((t )/(2+2 (t^2))),{t,0,2n-1}]

FullSimplify[%,n∈Integers&&n>0]

Table[Integrate[-(1/2) (((-1))^n)(t^(2n-1))Csch[π t],{t,0,∞}],{n,10}]

Table[-(((-1))^n) (-1+(4^n)) (((2 π))^(-2 n)) Gamma[2 n] Zeta[2 n],{n,10}]

Plus@@%//N

Integrate[-(1/2) (((-1))^n)Csch[π t](t^(2n-1)),{t,0,∞},Assumptions->n>1]

Sum[-(((-1))^n) (-1+(4^n)) (((2 π))^(-2 n)) Gamma[2 n] Zeta[2 n],{n,∞}]

SymbolicSum`SymbolicSum[-(((-1))^n) (-1+(4^n)) (((2 π))^(-2 n)) Gamma[2 n] Zeta[2 n],{n,∞}]

Series[((t Csch[π t])/(2+2 (t^2))),{t,0,8}]

Table[-(-1)^n π^(2n-1)BernoulliB[2n]((2^(2n-1))-1)/(2n)!,{n,0,5}]

Sum[-(-1)^n π^(2n-1)BernoulliB[2n]((2^(2n-1))-1)/(2n)!,{n,0,2k}]

Integrate[-(-1)^n π^(2n-1)BernoulliB[2n]((2^(2n-1))-1)/(2n)!(t^(2k)),{t,0,∞}]

Integrate[-(-1)^n π^(2n-1)BernoulliB[2n]((2^(2n-1))-1)/(2n)!(t^(2k)),{t,0,∞}]

SeriesTerm[((t Csch[t])/(2+2 (t^2))),{t,0,n}]

Series[((x Csch[(π/x)])/(2+2 (x^2)))(1/(x^2)),{x,∞,10}]

Series[(1/(x(1+(x^2)))),{x,0,5}]

SeriesTerm[(1/(x(1+(x^2)))),{x,0,2n-1}]

FullSimplify[(I^(2n)),n∈Integers]

(1/2)Integrate[(((-1))^n)(x^(2n-1))Csch[(π/x)],{x,0,∞}]

Integrate[(((-1))^n)(x^(2n-1))Csch[(π/x)],x]

Series[((t Csch[π t])/(2 (1+(t^2)))),{t,0,2}]

Related Integrals from Same Paper

(∫_0^∞)((sin((( t)/2)))/((e^(π t))-1))dt==(1/2) coth((m/2))-(1/m)==(1/((e^m)-1))-(1/m)+(1/2)

Integrate[((Sin[(t/2)])/(Exp[π t]-1)),{t,0,∞}]//Timing

(∫_0^∞)((sin(((m t)/2)))/((e^(π t))-1))dt==(1/2) coth((m/2))-(1/m)==(1/((e^m)-1))-(1/m)+(1/2)

Integrate[((Sin[((m t)/2)])/(Exp[π t]-1)),{t,0,∞}]//Timing

FullSimplify[Subtract@@{%,(1/(Exp[m]-1))-(1/m)+(1/2)}]

(∫_0^∞)((sin(m t))/((e^(2 π t))-1))dt==(((e^m)+1)/(4 ((e^m)-1)))-(1/(2 m))

Integrate[((Sin[m t])/(Exp[2π t]-1)),{t,0,∞}]//Timing

FullSimplify[Subtract@@{Last[%],(1/4)((Exp[m]+1)/(Exp[m]-1))-(1/(2m))}]

(∫_0^∞)(((e^(a t))-(e^(-a t)))/((e^(π t))-1))dt==(1/a)-cot(a)

Integrate[((Exp[a t]-Exp[-a t])/(Exp[π t]-1)),{t,0,∞}]//Timing

(∫_0^∞)(((e^(c t))-(e^(-c t)))/((e^(π t))-(e^(-π t))))dt==(1/2) tan((c/2))

Integrate[((Exp[c t]-Exp[-c t])/(Exp[π t]-Exp[-π t])),{t,0,∞}]//Timing