Inverse Cotangent

Author

Eric W. Weisstein
August 24, 2008

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

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

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

Definition

FullSimplify[TrigToExp[ArcCot[z]]]

FullSimplify[{ArcCot[1/z]-ArcTan[z],ArcCot[z]-ArcTan[1/z]}]

Alternate Definition

arcCot[z_]:=(1/2)I(Log[-I+z]-Log[I+z])

FullSimplify[arcCot[1/z]-ArcTan[z],z>0]

FullSimplify[arcCot[1/z]-ArcTan[z],z<0]

Plots

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

Branch Cut

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

Real Axis

PlotDiscontinuous[ArcCot[x],{x,-10,0,10},AxesLabel->TraditionalForm/@{x,ArcCot[x]},
HorizontalAsymptotes->{0},
Epilog->{PointSize[.025],Red,Point[{0,ArcCot[0]}],
Point[{0,-Pi/2}],{White,PointSize[.018],Point[{0,-Pi/2}]}},
Ticks->{Automatic,{#,TraditionalForm[#]}&/@Range[-Pi/2,Pi/2,Pi/4]}]

Real Axis Alternate Definition

PlotDiscontinuous[arcCot[z],{z,-10,10},PlotStyle->Red,HorizontalAsymptotes->{0,Pi},
AxesLabel->TraditionalForm/@{x,ArcCot[x]},
Ticks->{Automatic,{#,TraditionalForm[#]}&/@Range[0,Pi,Pi/2]},
PlotRange->{-.2,3.2}]

Complex Plane

ComplexPlot3D[ArcCot[z],{z,-5,5},PlotPoints->100]

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

Complex Plane Alternate Definition

ComplexPlot3D[arcCot[z],{z,-5,5},PlotPoints->100,ComplexPlotLabel->ArcCot[z],PlotRange->All]//Timing

ComplexContourPlot[arcCot[z],{z,-5,5},ComplexPlotLabel->ArcCot[z]]//Timing

Special Values

Limit[ArcCot[x],x->0,Direction->#]&/@{-1,1}

ArcCot[0]

Derivative and Integral

Derivative

D[ArcCot[z],z]

Integral

Integrate[ArcCot[z],z]

Series

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

V5

<<DiscreteMath`RSolve`

SeriesTerm[ArcCot[x],{x,0,n}]

Series[ArcCot[x],{x,∞,20}]

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

V7

SeriesCoefficient[ArcCot[x],{x,0,n}]//InputForm

Sum[% x^n,{n,0,∞}]//InputForm

Sums

(∑_n^∞)(((x^n) ((i^(n-1)) θ(n-1) (Subscript[δ, (n-1) mod 2])))/n)==(tan^(-1))(x)

Sum[(((I^(-1+n)) KroneckerDelta[Mod[-1+n,2]] UnitStep[-1+n])/n)x^n,{n,∞}]

Sum[(((I^(-1+n)) KroneckerDelta[Mod[-1+n,2]] UnitStep[-1+n])/n)x^n,{n,∞}]

FullSimplify[%30]

FunctionExpand[%]

FullSimplify[%]

Subtract@@@Table[{x Hypergeometric2F1[(1/4),1,(5/4),(x^4)]-(1/3) (x^3) Hypergeometric2F1[1,(3/4),(7/4),(x^4)],ArcTan[x]}/.x->RandomComplex[],{100}]//Chop//Union

Sum[(((I^(-1+n)) KroneckerDelta[Mod[-1+n,2]] UnitStep[-1+n])/n)x^n,{n,100}]

(∑_(n=0)^∞)(x^n) Piecewise[(({{(π/2), n==0}, {-((i^(n-1))/n), n mod 2==1∧n≥0}})),0]==(cot^(-1))(x)

SeriesCoefficient[ArcCot[x],{x,0,n}]//InputForm

Sum[% x^n,{n,0,∞}]//InputForm

(π/2)-(∑_(k=0)^∞)(((((-1))^k) (x^(2 k+1)))/(2 k+1))==(cot^(-1))(x)

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

FullSimplify[%-ArcCot[x]]

Table[(1/2) (π-2 ArcCot[x]-2 ArcTan[x])/.x->RandomComplex[],{100}]//Chop//Union

(∑_(k=0)^∞)(((((-1))^k) (z^(-(2 k+1))))/(2 k+1))==(cot^(-1))(z)

Sum[(((((-1))^k)(z^(-(2k+1))))/(2k+1)),{k,0,∞}]

FullSimplify[%]

FunctionExpand[%]

z (∑_n^∞)(((2 n-2)!!)/((2 n-1)!! ((((z^2)+1))^n)))==(cot^(-1))(z)

FullSimplify[(((2n-2)!!)/((2n-1)!!((((z^2)+1))^n)))]

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

{((z ArcSin[(1/(Sqrt[1+(z^2)]))])/(Sqrt[(z^2)])),ArcCot[z]}/.z->-0.01`2*^-30 I//Chop

z Sum[(((2n-2)!!)/((2n-1)!!((((z^2)+1))^n))),{n,∞},GenerateConditions->True]

FullSimplify[%]

With[{z=-0.01`20 I},z Sum[(((2n-2)!!)/((2n-1)!!((((z^2)+1))^n))),{n,5000}]]

ArcCot[z]/.z->-0.01`20 I

((z ArcSin[(1/(Sqrt[1+(z^2)]))])/(Sqrt[(z^2)]))/.z->-0.01`20 I

FullSimplify[%,z>0]

FullSimplify[%-ArcCot[z],z>0]

Table[-ArcCot[z]+ArcCsc[(Sqrt[1+(z^2)])]/.z->RandomReal[{0,10}],{1000}]//Chop//Union

Subtract@@@Table[{((z ArcCsc[(Sqrt[1+(z^2)])])/(Sqrt[(z^2)])),ArcCot[z]}/.z->RandomComplex[],{1000}]//Chop//Union

(∑_n^∞)(cot^(-1))((n^2))==(cot^(-1))(((-1-cot((π/(Sqrt[2]))) tanh((π/(Sqrt[2]))))/(cot((π/(Sqrt[2]))) tanh((π/(Sqrt[2])))-1)))

V6

Sum[ArcCot[n^2],{n,Infinity}]

V7

Sum[ArcCot[n^2],{n,Infinity}]

FullSimplify[%]

Identities

<<Utilities`Plot`
<<Utilities`Simplify`

2 (cot^(-1))(2 x)-(cot^(-1))(x)==(cot^(-1))(4 (x^3)+3 x)

TraditionalForm[id=2ArcCot[2x]-ArcCot[x]-ArcCot[4x^3+3x]]

ComplexPlot3D[id,{x,-5,5},ComplexPlotLabel->"identity 1",PlotPoints->100,PlotRange->All]

FullSimplify[id,x∈Reals]

Log[FullSimplify[TrigSimplify[PowerExpand[FullSimplify[TrigSimplify[Exp[id-Pi]]]]]]]

-(cot^(-1))(x)+3 (cot^(-1))(3 x)==(cot^(-1))(((27 (x^4)+18 (x^2)-1)/(8 x)))

TraditionalForm[id2=3ArcCot[3x]-ArcCot[x]-ArcCot[(27x^4+18x^2-1)/(8x)]
]

{Sqrt[-1/3+2/(3*Sqrt[3])],-Sqrt[-1/3+2/(3*Sqrt[3])]}//N

ComplexPlot3D[Abs[id2],{x,-.5,.5},ComplexPlotLabel->"identity 2",PlotRange->All,
PlotPoints->150]

ComplexContourPlot[id2,{x,-.5,.5},ComplexPlotLabel->"identity 2",PlotRange->All,PlotPoints->150,ColorFunction->(Hue[#/2]&)]

Plot[3ArcCot[3x]-ArcCot[x]-ArcCot[(27x^4+18x^2-1)/(8x)],{x,-.5,.5},PlotStyle->Hue[0],PlotRange->All]

Log[FullSimplify[TrigSimplify[PowerExpand[FullSimplify[Exp[TrigSimplify[id2]]]]]]]

Valid for all z

ArcCot[0]

{ArcTan[Infinity],ArcTan[1/0]}

FullSimplify[ArcTan[1/z]]

FullSimplify[-ArcCot[-z]]

FullSimplify[-Pi+ArcSec[Sqrt[x^2+1]/x],#]&/@{x<0,x>0}

Plot[If[x<0,-Pi+ArcSec[Sqrt[x^2+1]/x],ArcSec[Sqrt[x^2+1]/x]]-ArcCot[x],{x,-5,5},PlotStyle->Red]

FullSimplify[Pi/2-ArcTan[x],#]&/@{x<0,x>0}

Plot[If[x<0,-Pi/2-ArcTan[x],Pi/2-ArcTan[x]]-ArcCot[x],{x,-5,5},PlotStyle->Red]

FullSimplify[ArcSin[1/Sqrt[x^2+1]],#]&/@{x<0,x>0}

Plot[If[x<0,-ArcSin[1/Sqrt[x^2+1]],ArcSin[1/Sqrt[x^2+1]]]-ArcCot[x],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[-Pi/2-ArcCot[1/x],#]&/@{x<0,x>0}

Plot[If[x<0,-Pi/2-ArcCot[1/x],Pi/2-ArcCot[1/x]]-ArcCot[x],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[-ArcCsc[Sqrt[x^2+1]],#]&/@{x<0,x>0}

Plot[If[x<0,-ArcCsc[Sqrt[x^2+1]],ArcCsc[Sqrt[x^2+1]]]-ArcCot[x],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[ArcCos[x/Sqrt[x^2+1]],#]&/@{x<0,x>0}

Plot[If[x<0,-Pi+ArcCos[x/Sqrt[x^2+1]],ArcCos[x/Sqrt[x^2+1]]]-ArcCot[x],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[Pi/2-ArcSin[x/Sqrt[x^2+1]],#]&/@{x<0,x>0}

Plot[If[x<0,-Pi/2,Pi/2]-ArcSin[x/Sqrt[x^2+1]]-ArcCot[x],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[Pi/2+ArcTan[-x],#]&/@{x<0,x>0}

Plot[If[x<0,-Pi/2,Pi/2]+ArcTan[-x]-ArcCot[x],{x,-5,5},PlotStyle->Red,PlotRange->All]