Inverse Hyperbolic Cosine

Author

Eric W. Weisstein
July 17, 2014

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

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

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

Definition

TrigToExp[ArcCosh[z]]

Simplify[TrigToExp[ArcCosh[z]],z>1]

Simplify[TrigToExp[ArcCosh[z]],z<-1]

Gradshteyn and Ryzhik, corr. enl. ed., pp. xxvii and xxviii

Claimed:

ArcCosh[z]==I ArcCos[z]

FullSimplify[ArcCosh[z]-I ArcCos[z]]

DensityPlot[#[ArcCosh[x+I y]-I ArcCos[x+I y]],{x,-10,10},{y,-10,10}]&/@{Re,Im}

DensityPlot[#[ArcCosh[x+I y]+I ArcCos[x+I y]],{x,-10,10},{y,-10,10}]&/@{Re,Im}

Actual:

ArcCosh[z]==Sign[Im[z]]I ArcCos[z]/;Im[z]≠0

FullSimplify[ArcCosh[z]-Sign[Im[z]]I ArcCos[z]]

Table[ArcCosh[z]-I ArcCos[z] Sign[Im[z]]/.z->RandomComplex[10{-1,1}(I+1)],{1000}]//Chop//Union

FindInstance[ArcCosh[z]-I ArcCos[z] Sign[Im[z]]≠0,z,Complexes]

ArcCosh[z]-Sign[Im[z]]I ArcCos[z]/.z->0

FindInstance[ArcCosh[z]-I ArcCos[z] Sign[Im[z]]≠0&&z≠0,z,Complexes]

FindInstance[ArcCosh[z]-I ArcCos[z] Sign[Im[z]]≠0&&Im[z]≠0,z,Complexes]

FullSimplify[ArcCosh[z]-I ArcCos[z] Sign[Im[z]],Im[z]≠0]

Plot[#[ArcCosh[x]-I ArcCos[x]],{x,-10,10}]&/@{Re,Im}

Plot[#[ArcCosh[x]-I ArcCos[x]],{x,-10,10}]&/@{Re,Im}

Messages are bug(277606)

DensityPlot[Evaluate[#[ArcCosh[x+I y]-Sign[y]I ArcCos[x+I y]]],{x,-10,10},{y,-10,10},WorkingPrecision->30]&/@{Re,Im}

Plots

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

Branch Cuts

BranchCuts[ArcCosh]

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

Real Axis

Plot[ArcCosh[x],{x,1,10},
PlotStyle->Red,Epilog->{Red,PointSize[.02],Point[{1,0}]},AxesLabel->TraditionalForm/@{x,ArcCosh[x]}]

ComplexPlot3D[ArcCosh[z],{z,-5,5},PlotPoints->50,Mesh->False,ClipFill->None]//Timing

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

Derivative and Integral

Derivative

D[ArcCosh[z],z]

Integral

Integrate[ArcCosh[z],z]

Special Values

ArcCosh[0]

ArcCosh/@{1,∞}

Series

About 0

Series[ArcCosh[x],{x,0,10}]

V5

<<DiscreteMath`RSolve`

SeriesTerm[ArcCosh[z],{z,0,n}]

Sum[I (-1)^((n+1)/2)LegendreP[-1+n,0]/n x^n,{n,10}]

V7

SeriesCoefficient[ArcCosh[z],{z,0,n}]

About 1

Series[ArcCosh[x],{x,1,5}]

Series[ArcCosh[x],{x,1,5}]/Sqrt[2]/Sqrt[x-1]

SeriesTerm[ArcCosh[x],{x,1,n}]

Take[l=CoefficientList[Series[ArcCosh[x],{x,1,20}]/Sqrt[2]/Sqrt[x-1]/.x->z+1,z],10]

Through[{Numerator,Denominator}[l]]

About ∞

Series[ArcCosh[1/x],{x,0,6}]

l=Rest[First/@Partition[CoefficientList[Series[ArcCosh[1/x],{x,0,20}],x],2]]

Through[{Numerator,Denominator}[l]]

Series[ArcCosh[x],{x,Infinity,6}]

V5

<<DiscreteMath`RSolve`

SeriesTerm[ArcCosh[1/x],{x,0,n}]

SeriesTerm[ArcCosh[x],{x,Infinity,n}]

V7

SeriesCoefficient[ArcCosh[1/x],{x,0,n}]

SeriesCoefficient[ArcCosh[x],{x,∞,n}]

SeriesCoefficient[ArcCosh[x]+Log[1/x],{x,∞,n}]

Series[ArcCosh[x]+Log[1/x],{x,∞,17}]//Normal

Series[ArcCosh[1/x],{x,0,17}]//Normal

SeriesCoefficient[ArcCosh[1/x],{x,0,n}]

-Table[(((2n-1)!!)/(2n(2n)!!))(x^(-2n)),{n,8}]

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

FullSimplify[%]

FullSimplify[%-ArcCosh[x]]

Series[%+Log[1/x],{x,∞,10}]

Subtract@@@Table[{ArcCosh[x],Log[1+(Sqrt[1-(1/(x^2))])]-Log[(1/x)]}/.x->RandomComplex[],{1000}]//Chop//Union

Sums

(∑_n^∞)((i (((-1))^((n+1)/2)) (Subscript[P, n-1])(0) (x^n))/n)+((i π)/2)==(cosh^(-1))(x)

Series[ArcCosh[x],{x,0,10},Assumptions->x∈Reals]

((I π)/2) + Sum[I (-1)^((n+1)/2)LegendreP[-1+n,0]/n x^n,{n,10}]

((I π)/2) + Sum[I (-1)^((n+1)/2)LegendreP[-1+n,0]/n x^n,{n,∞}]//InputForm

((I π)/2) + Sum[I (-1)^((n+1)/2)LegendreP[-1+n,0]/n x^n,{n,20}]/.x->.6

ArcCosh[.6]

((π i)/2)-i (∑_(n=0)^∞)(((Subscript[((1/2)), n]) (x^(2 n+1)))/((2 n+1) n!))==(cosh^(-1))(x)

Series[ArcCosh[x],{x,0,20}]

Series[ArcCosh[x],{x,0,20},Assumptions->x∈Reals]

ser=CoefficientList[%,x]

ser[[2;;-1;;2]]I

Through[{Numerator,Denominator}[%]]

(π/2)I-I Sum[((Pochhammer[1/2,n])/(n!(2n+1)))(x^(2n+1)),{n,0,4}]

(π/2)I-I Sum[((Pochhammer[1/2,n])/(n!(2n+1)))(x^(2n+1)),{n,0,∞}]

FullSimplify[%]

{I ArcCos[.9],ArcCosh[.9]}

{I ArcCos[#],ArcCosh[#]}&/@{.9,1.1}//Chop

FullSimplify[I ArcCos[x]-ArcCosh[I x]]

(∑_(n=0)^∞)(x^n) Piecewise[(({{((i π)/2), n==0}, {-((i (Subscript[((1/2)), ((n-1)/2)]))/(n ((n-1)/2)!)), n mod 2==1∧n≥1}, {0, "PiecewiseDefault"[True]}}))]==(cosh^(-1))(x)

Sum[(x^n)(Piecewise[{{((I π)/2), n==0}, {-((I Pochhammer[(1/2),(1/2) (-1+n)])/(n ((1/2) (-1+n))!)), Mod[n,2]==1&&n≥1}, {0, True}}]),{n,0,∞}]

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

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

FullSimplify[Log[2]-Log[(1/x)]-((Log[(1/2) (1+(Sqrt[((-1+(x^2))/(x^2))]))])/((x^2) (-1+(Sqrt[((-1+(x^2))/(x^2))])) (1+(Sqrt[((-1+(x^2))/(x^2))]))))]

{Log[1+(Sqrt[1-(1/(x^2))])]-Log[(1/x)],ArcCos[x]}/.x->-1`30//Chop

FullSimplify[%,x>0]

FullSimplify[%-ArcCosh[x],x>0]

Series[%+Log[1/x],{x,∞,10}]

Subtract@@@Table[{ArcCosh[x],Log[1+(Sqrt[1-(1/(x^2))])]-Log[(1/x)]}/.x->RandomComplex[],{1000}]//Chop//Union