Natural Logarithm

Author

Eric W. Weisstein
March 20, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Branch Cut

Branch Cuts

BranchCuts[Log]//TraditionalForm

With[{f=Log},
BranchCutPlot[f,Epilog->Text[TraditionalForm[f[z]],{-2,2.3}]]
]

Definitions of Inverse Trigs and Hyperbolics

{#[z],Simplify[TrigToExp[#[z]]]}&/@{ArcCsc,ArcCos,ArcCot,ArcCsch,ArcCosh,ArcCoth,ArcSech,ArcSinh,ArcTanh,ArcSec,ArcSin,ArcTan}//TableForm//TraditionalForm

Integral Plot

<<Graphics`InequalityGraphics`
<<Utilities`Typesetting`

Show[Block[{$DisplayFunction=Identity},
{
Plot[1/x,{x,.001,3},
Epilog->{Dashing[{.02}],Line[{{1,0},{1,1}}],Line[{{E,0},{E,1/E}}]}],
InequalityPlot[1<x<E&&0<y<1/x,{x,0,3},{y,0,2}]
}],AspectRatio->Automatic,PlotRange->{{0,3},{0,2}},
Ticks->{{1,2,{E,TraditionalForm[E]},3},
{#,NumberForm[#,{3,2},NumberPadding->{"","0"}]}&/@{0.5,1.0,1.5,2.}
},
AxesLabel->TraditionalForm/@{x,y==1/x}]

Plots

<<Utilities`Plot`

Real Axis

Plot[Log[x],{x,.001,10},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,Log[x]}]

Complex Plane

ComplexPlot3D[Log[z],{z,-5,5},PlotPoints->100]//Timing

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

Integral Definition

This returns a condition that is not the tightest possible

Integrate[1/t,{t,1,x}]

Integrate[1/t,{t,1,x},Assumptions->x>1]

ln x is defined for x>0

Integrate[1/t,{t,1,x},Assumptions->0<x<1]

This automatically takes the principal value

Integrate[1/t,{t,1,x},Assumptions->x<0,PrincipalValue->False]

Integrate[1/t,{t,1,x},Assumptions->x<0]

This is wrong; the integral is defined only in the principal value sense

Integrate[1/t,{t,1,x},Assumptions->x<0,PrincipalValue->False]

Limit

Limit[(1+1/x)^x,x->∞]

Special Values

Log[1]

Log[0]

Log[-1]

{Log[I],Log[-I]}

Continued Fractions

(x/(ContinuedFractionK[x ((⌈(n/2)⌉)^2),n+1,{n,∞}]+1))==ln(x+1)

x/(1+ContinuedFractionK[Ceiling[n/2]^2x,n+1,{n,∞}])

x/(1+ContinuedFractionK[Ceiling[n/2]^2x,n+1,{n,10}])/.x->.3

Log[1+.3]

((2 x)/(ContinuedFractionK[-(n^2) (x^2),2 n+1,{n,∞}]+1))==ln(((x+1)/(1-x)))

2x/(1+ContinuedFractionK[-n^2x^2,(2n+1),{n,∞}])

2x/(1+ContinuedFractionK[-n^2x^2,(2n+1),{n,20}])/.x->.3

Log[(1+.3)/(1-.3)]

Series

About -1

Series[Log[1+x],{x,0,10}]

SeriesCoefficient[Log[1+x],{x,0,n}]

Sums

(∑_n^∞)(((((-1))^(n+1)) (x^n))/n)==ln(x+1)

Sum[((((-1))^(n+1))/n)(x^n),{n,∞}]

(∑_n^∞)((2 (x^(2 n-1)))/(2 n-1))==ln(((x+1)/(1-x)))

Sum[(2/(2n-1))(x^(2n-1)),{n,∞}]

FullSimplify[%-Log[((1+x)/(1-x))]]

2 ArcTanh[x]-Log[((1+x)/(1-x))]/.x->.9

6 (∑_n^∞)(1/(n ((e^(π n))-1)))-4 (∑_n^∞)(1/(n ((e^(2 π n))-1)))+10 (∑_n^∞)(1/(n ((e^(π n))+1)))==log(2)

10Sum[(1/(n((E^(π n))+1))),{n,∞}]+6Sum[(1/(n((E^(π n))-1))),{n,∞}]-4Sum[(1/(n((E^(2π n))-1))),{n,∞}]//Timing

With[{nmax=∞},10NSum[(1/(n((E^(π n))+1))),{n,nmax}]+6NSum[(1/(n((E^(π n))-1))),{n,nmax}]-4NSum[(1/(n((E^(2π n))-1))),{n,nmax}]]//N[#,20]&

9 (∑_n^∞)(1/(n ((e^(π n))-1)))+(∑_n^∞)(1/(n ((e^(3 π n))-1)))+(49/3) (∑_n^∞)(1/(n ((e^(π n))+1)))-(14/3) (∑_n^∞)(1/(n ((e^(2 π n))+1)))-(7/3) (∑_n^∞)(1/(n ((e^(3 π n))+1)))+(2/3) (∑_n^∞)(1/(n ((e^(6 π n))+1)))==log(3)

9Sum[(1/(n((E^(π n))-1))),{n,∞}]+(49/3)Sum[(1/(n((E^(π n))+1))),{n,∞}]-(14/3)Sum[(1/(n((E^(2π n))+1))),{n,∞}]+Sum[(1/(n((E^(3π n))-1))),{n,∞}]-(7/3)Sum[(1/(n((E^(3π n))+1))),{n,∞}]+(2/3)Sum[(1/(n((E^(6π n))+1))),{n,∞}]//Timing

9Sum[(1/(n((E^(π n))-1))),{n,∞}]+(49/3)Sum[(1/(n((E^(π n))+1))),{n,∞}]-(14/3)Sum[(1/(n((E^(2π n))+1))),{n,∞}]+Sum[(1/(n((E^(3π n))-1))),{n,∞}]-(7/3)Sum[(1/(n((E^(3π n))+1))),{n,∞}]+(2/3)Sum[(1/(n((E^(6π n))+1))),{n,∞}]//N[#,20]&

Log[3]//N[#,20]&

(57/4) (∑_n^∞)(1/(n ((e^(π n))-1)))+(3/4) (∑_n^∞)(1/(n ((e^(5 π n))-1)))+(91/4) (∑_n^∞)(1/(n ((e^(π n))+1)))-(13/2) (∑_n^∞)(1/(n ((e^(2 π n))+1)))-(7/4) (∑_n^∞)(1/(n ((e^(5 π n))+1)))+(1/2) (∑_n^∞)(1/(n ((e^(10 π n))+1)))==log(5)

(57/4)Sum[(1/(n((E^(π n))-1))),{n,∞}]+(91/4)Sum[(1/(n((E^(π n))+1))),{n,∞}]-(13/2)Sum[(1/(n((E^(2π n))+1))),{n,∞}]+(3/4)Sum[(1/(n((E^(5π n))-1))),{n,∞}]-(7/4)Sum[(1/(n((E^(5π n))+1))),{n,∞}]+(1/2)Sum[(1/(n((E^(10π n))+1))),{n,∞}]//Timing

%//N[#,20]&

Log[5]//N[#,20]&

Renormalized Sums

(∑_k^∞)(((-1))^(k+1)) ln(k)

V6

FullSimplify[Sum[(-1)^(k+1)Log[k],{k,Infinity}]]

V7

Sum[(-1)^(k+1)Log[k],{k,Infinity}]

Sum[(-1)^(k+1)Log[k],{k,Infinity},Regularization->"Dirichlet"]

FullSimplify[%]

(∑_k^∞)ln(k)==(1/2) ln(2 π)

V6

FullSimplify[Sum[Log[k],{k,Infinity}]]

V7

Sum[Log[k],{k,Infinity}]

Sum[Log[k],{k,Infinity},Regularization->"Dirichlet"]

Identities

Equality of log(x+(Sqrt[(x^2)-1])) and-log(x-(Sqrt[(x^2)-1]))

f[x_]:=Log[x+Sqrt[x^2-1]]+Log[x-Sqrt[x^2-1]]

Chop@Table[f[Random[Complex,{-1,1}2(I+1)]],{20}]