Logarithmic Integral

Author

Eric W. Weisstein
August 17, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`

Along Real Axis

PlotDiscontinuous[LogIntegral[x],{x,0,1,10},
AxesLabel->TraditionalForm/@{x,LogIntegral[x]},
VerticalAsymptotes->{1}
]

Complex Plane

ComplexPlot3D[LogIntegral[z],{z,-5,5}]//Timing

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

Soldner's Constant

V6

soldner=x/.FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->200]

Integrate[1/Log[u],{u,sold,x},Assumptions->1<sold<x]

FullSimplify[%]

Chop[%/.sold->soldner]

V7

x/.FindInstance[LogIntegral[x]==0&&x>0,x][[1]]

N[%,200]

Derivative

D[LogIntegral[x],x]

Integrals

Indefinite

Integrate[LogIntegral[x],x]

TraditionalForm[%]

Definite

Integrate[LogIntegral[x],{x,0,1}]

N[%]

Integrate[LogIntegral[x],{x,1,∞}]

Special Values

LogIntegral[0]

LogIntegral[1]

LogIntegral[soldner]

LogIntegral[2]

N[%,20]

Definition

v4.2

Integrate[1/Log[u],{u,0,x},Assumptions->x>0]

Integrate[1/Log[u],{u,0,x},PrincipalValue->True]

v5.0

Integrate[1/Log[u],{u,0,x},Assumptions->x>0]

Integrate[1/Log[u],{u,0,x},PrincipalValue->True]

v6.0

Integrate[1/Log[u],{u,0,x}]

FullSimplify[%,0<x<1]

Integrate[1/Log[u],{u,0,x},Assumptions->x>0]

FullSimplify[%,0<x<1]

Integrate[1/Log[u],{u,0,x},PrincipalValue->True]

FullSimplify[%,x∈Reals]

Series

s=Series[LogIntegral[z],{z,1,10}]

l=Rest[CoefficientList[Series[LogIntegral[z],{z,1,10}]/.z->1-x,x]]//InputForm

Through[{Numerator,Denominator}[%]]

Similar to A002206

V6

Sum[BernoulliB[j]/j StirlingS1[n,j-1],{j,n+1}]/n!

Series[1/Log[x+1],{x,0,15}]

numer=Table[%,{n,9}]

Drop[l,2]/numer

Table[-Sum[BernoulliB[j]/j StirlingS1[n,j-1],{j,n+1}](-1)^(n+1)/ (n+1)!,{n,0,9}]//InputForm

V7

Sum[BernoulliB[j]/j StirlingS1[n,j-1],{j,n+1}]/n!

Li(n)

FullSimplify[Integrate[1/Log[u],{u,2,x}]]

N[LogIntegral[2],20]

Identities

(∑_n^∞)(((ln^n)(x))/(n n! (m^n)))-ln(m)+ln(ln(x))+ℽ

Bromwich and Hardy

V6

EulerGamma+Log[Log[x]]-Log[m]+Sum[Log[x]^n/n/n!/m^n,{n,Infinity}]

FullSimplify[%,x>1&&m>0]

FullSimplify[%-LogIntegral[x^(1/m)],x>1&&m>0]

V7

EulerGamma+Log[Log[x]]-Log[m]+Sum[Log[x]^n/n/n!/m^n,{n,Infinity}]

FullSimplify[%,x>1&&m>0]

FullSimplify[%-LogIntegral[x^(1/m)],x>1&&m>0]

EulerGamma+Log[Log[x]]-Log[m]+Sum[Log[x]^n/n/n!/m^n,{n,Infinity},Assumptions->x>1&&m>0]

(∑_k^∞)(((ln^k)(x))/(k k!))+ln(ln(x))+ℽ

Nielsen and Ramanujan

V6

FullSimplify[EulerGamma+Log[Log[x]]+Sum[Log[x]^k/k/k!,{k,∞}],x>1]

V7

FullSimplify[EulerGamma+Log[Log[x]]+Sum[Log[x]^k/k/k!,{k,∞}],x>1]

(Sqrt[x]) (∑_(n=0)^∞)(∑_(k=0)^(⌊((n-1)/2)⌋))(((((-1))^(n-1)) (ln^n)(x))/((2 k+1) (2^(n-1)) n!))+ln(ln(x))+ℽ

Ramanujan

V6

EulerGamma+Log[Log[x]]+Sqrt[x]Sum[(-1)^(n-1)Log[x]^n/n!/2^(n-1) 1/(2k+1),{n,0,∞},{k,0,Floor[(n-1)/2]}]

FullSimplify[Sum[1/(2k+1),{k,0,Floor[(n-1)/2]}],n∈Integers]

Refine[%,Mod[n,2]==#]&/@{0,1}//Simplify

FullSimplify[EulerGamma+Log[Log[x]]+Sqrt[x](
Sum[(((((-1))^(n-1)) ((Log[x])^n))/((n! (2^(n-1))) ))((1/2) HarmonicNumber[(n-1)/2]+Log[2]),{n,0,∞,2}]+Sum[(((((-1))^(n-1)) ((Log[x])^n))/((n! (2^(n-1))) ))((1/2) HarmonicNumber[n/2]+Log[2]),{n,1,∞,2}]
),x>1]

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

V7

TimeConstrained[EulerGamma+Log[Log[x]]+Sqrt[x]Sum[(-1)^(n-1)Log[x]^n/n!/2^(n-1) 1/(2k+1),{n,0,∞},{k,0,Floor[(n-1)/2]}],60]//Timing

FullSimplify[Sum[1/(2k+1),{k,0,Floor[(n-1)/2]}],n∈Integers]

Refine[%,Mod[n,2]==#]&/@{0,1}//Simplify

TimeConstrained[FullSimplify[EulerGamma+Log[Log[x]]+Sqrt[x](
Sum[(((((-1))^(n-1)) ((Log[x])^n))/((n! (2^(n-1))) ))((1/2) HarmonicNumber[(n-1)/2]+Log[2]),{n,0,∞,2}]+Sum[(((((-1))^(n-1)) ((Log[x])^n))/((n! (2^(n-1))) ))((1/2) HarmonicNumber[n/2]+Log[2]),{n,1,∞,2}]
),x>1],60]//Timing

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