Inverse Tangent

Author

Eric W. Weisstein
December 5, 2016

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

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

©2016 Wolfram Research, Inc. except for portions noted otherwise

Definition

Simplify[TrigToExp[ArcTan[x]]]

Plots

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

Branch Cuts

BranchCuts[ArcTan]//TraditionalForm

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

Real Axis

PlotDiscontinuous[ArcTan[x],
{x,-10,10},AxesLabel->TraditionalForm/@{x,ArcTan[x]},
HorizontalAsymptotes->{-1,1}Pi/2,
PlotStyle->Red,
PlotRange->{Automatic,{-1.8,1.8}},
Ticks->{Automatic,{#,TraditionalForm[#]}&/@Range[-Pi/2,Pi/2,Pi/4]}]

Complex Plane

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

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

Ratio of Reals

ComplexPlot3D[ArcTan[Im[z]/Re[z]],{z,-5,5},PlotPoints->100,ComplexPlotLabel->ArcTan[y/x]]//Timing

ComplexContourPlot[ArcTan[Im[z]/Re[z]],{z,-5,5},ComplexPlotLabel->ArcTan[y/x]]//Timing

Two-Argument

ComplexPlot3D[ArcTan[Im[z],Re[z]],{z,-5,5},PlotPoints->100,ComplexPlotLabel->ArcTan[x,y]]//Timing

ComplexContourPlot[ArcTan[Im[z],Re[z]],{z,-5,5},ComplexPlotLabel->ArcTan[x,y]]//Timing

Special Values

ArcTan/@{-∞,-I,0,I,∞}

Derivative and Integral

Derivative

D[ArcTan[z],z]

Integral

Integrate[ArcTan[z],z]

Identities

In Terms of Hypergeometrics

z Hypergeometric2F1[1,1/2,3/2,-z^2]

FullSimplify[x/(1+x^2) Hypergeometric2F1[1,1,3/2,x^2/(1+x^2)],x∈Reals]

Plot[ArcSin[(x/(Sqrt[1+(x^2)]))]-ArcTan[x],{x,-5,5},PlotStyle->Red]

In Terms of Itself

FullSimplify[-ArcTan[-z]]

FullSimplify[ArcCot[1/z]]

For Real x

FullSimplify[-Pi/2-ArcTan[1/x],x<0]

Plot[Switch[z,
_?Negative,-Pi/2-ArcTan[1/x]-ArcTan[x],
_?Positive,Pi/2-ArcTan[1/x]-ArcTan[x]
],
{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[Pi/2-ArcCos[x/Sqrt[x^2+1]],x∈Reals]

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

FullSimplify[Pi/2-ArcCos[x/Sqrt[x^2+1]],x∈Reals]

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

FullSimplify[ArcCot[-x]-Pi/2,x<0]

FullSimplify[ArcCot[-x]+Pi/2,x>0]

Plot[Switch[x,
_?Negative,ArcCot[-x]-Pi/2-ArcTan[x],
_?Positive,ArcCot[-x]+Pi/2-ArcTan[x]
],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[-ArcCot[x]-Pi/2,x<0]

Plot[Switch[x,
_?Negative,-ArcCot[x]-Pi/2-ArcTan[x],
_?Positive,-ArcCot[x]+Pi/2-ArcTan[x]
],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[ArcSin[x/Sqrt[x^2+1]],x∈Reals]

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

FullSimplify[ArcCos[1/Sqrt[x^2+1]],x>0]

FullSimplify[-ArcCos[1/Sqrt[x^2+1]],x<0]

FullSimplify[-ArcCos[1/Sqrt[x^2+1]],x∈Reals]

FullSimplify[ArcCsc[Sqrt[x^2+1]/x],x∈Reals]

Plot[ArcCsc[Sqrt[x^2+1]/x]-ArcTan[x],{x,-5,5},PlotStyle->Red,PlotRange->All]

FullSimplify[ArcSec[Sqrt[x^2+1]],x∈Reals]

FullSimplify[-ArcSec[Sqrt[x^2+1]],x<0]

FullSimplify[ArcSec[Sqrt[x^2+1]],x>0]

(tan^(-1))(x)+(tan^(-1))(y)==(tan^(-1))(((x+y)/(1-x y)))

FullSimplify[Subtract@@{ArcTan[x]+ArcTan[y],ArcTan[(x+y)/(1-x y)]},-1<x<1&&-1<y<1]

Table[ArcTan[x]+ArcTan[y]-ArcTan[((x+y)/(1-x y))]/.Thread[{x,y}->RandomReal[{0,1},{2,1000}]]]//Chop//Union

Table[ArcTan[x]+ArcTan[y]-ArcTan[((x+y)/(1-x y))]/.Thread[{x,y}->RandomReal[{0,1.01},{2,10^4}]]]//Chop//Union

<<Utilities`Simplify`

?TrigSimplify

FullSimplify[TrigSimplify[ArcTan[x]+ArcTan[y]-ArcTan[(x+y)/(1-x y)]]]

(tan^(-1))((1/(a-b)))==(tan^(-1))((1/a))+(tan^(-1))((b/((a^2)-b a+1))) /; (a>b&&a>0)||(a<b&&a<0)

FullSimplify[Subtract@@{ArcTan[1/(a-b)],ArcTan[1/a]+ArcTan[b/(a^2-a b+1)]}]

Plot3D[ArcTan[1/(a-b)]-(ArcTan[1/a]+ArcTan[b/(a^2-a b+1)]),{a,-2,2},{b,-2,2
},PlotRange->1,AxesLabel->{x,y,z}]//Quiet

Graphics[Point[pts=Select[RandomReal[{-1,1},{10^4,2}],(#[[1]]>#[[2]]&&#[[1]]>0)||(#[[1]]<#[[2]]&&#[[1]]<0)&]],Axes->True]

Length[pts]

ListPointPlot3D[Function[{a,b},{a,b,Subtract@@{ArcTan[1/(a-b)],ArcTan[1/a]+ArcTan[b/(a^2-a b+1)]}}]@@@pts,PlotRange->1]

Function[{a,b},Subtract@@{ArcTan[1/(a-b)],ArcTan[1/a]+ArcTan[b/(a^2-a b+1)]}]@@@pts//Chop//Union

FullSimplify[Subtract@@{ArcTan[1/(a-b)],ArcTan[1/a]+ArcTan[b/(a^2-a b+1)]},(a>b&&a>0)||(a<b&&a<0)]

FullSimplify[TrigSimplify[ArcTan[1/(a-b)]-ArcTan[1/a]-ArcTan[b/(a^2-a b+1)]]]

q=FullSimplify[TrigSimplify[PowerExpand[FullSimplify[TrigSimplify[ArcTan[1/a]-2ArcTan[1/(2a)]+ArcTan[1/(4a^3+3a)]+Pi]]]]]

TrigSimplify[ArcTan[1/p]-ArcTan[1/(p+q)]-ArcTan[q/(p^2+p q+1)]]

(tan^(-1))((1/a))==2 (tan^(-1))((1/(2 a)))-(tan^(-1))((1/(4 (a^3)+3 a)))

FullSimplify[Subtract@@{ArcTan[(1/a)],2ArcTan[(1/(2a))]-ArcTan[(1/(4(a^3)+3a))]}]

Plot[Subtract@@{ArcTan[(1/a)],2ArcTan[(1/(2a))]-ArcTan[(1/(4(a^3)+3a))]},{a,-2,2}]

Plot3D[ReIm[Subtract@@{ArcTan[(1/(a+b I))],2ArcTan[(1/(2(a+b  I)))]-ArcTan[(1/(4(((a+b  I))^3)+3(a+b  I)))]}],{a,-4,4},{b,-4,4},PlotRange->.1,PlotPoints->50]

Function[a,Subtract@@{ArcTan[(1/a)],2ArcTan[(1/(2a))]-ArcTan[(1/(4(a^3)+3a))]}]/@RandomComplex[{-1,1}2(I+1),{10^4}]//Chop//Union

(tan^(-1))(p+r)+(tan^(-1))(p+q)-(tan^(-1))(p)=(π/2)

id=ArcTan[p+r]+ArcTan[p+q]-ArcTan[p]-Pi/2/.r->(1+p^2)/q

id=ArcTan[p+r]-Pi/2+ArcTan[p+q]-ArcTan[p]/.r->(1+p^2)/q

FullSimplify[id]

Table[id/.{p->Random[Complex,{-10,10}],q->Random[Complex,{-10,10}]},{20}]//Chop

Table[id/.{p->Random[Real,{-100,100}],q->Random[Real,{-100,100}]},{20}]//Chop

Table[id/.{p->Random[Real,{0,100}],q->Random[Real,{0,100}]},{20}]//Chop

FullSimplify[Together//@Exp[TrigToExp[id]],p>0&&q>0]

FullSimplify[ExpandAll[Pi+TrigToExp[id]]//.c_ Log[a_]:>Log[a^c]//.Log[a_]+Log[b_]:>Log[a b]//.a_^e_ b_^e_:>(a b)^e//.a_^e_ b_^f_:>(a/b)^e/;e==-f]

Series

About 0

V5

<<DiscreteMath`RSolve`

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

Table[-(1/2) π KroneckerDelta[n]+((Sin[((n π)/2)])/n),{n,10}]

series=Series[ArcTan[x],{x,0,20}]

coefs=Last/@Partition[CoefficientList[series,x],2]

Through[{Numerator,Denominator}[coefs]]

series=Series[ArcTan[1+x],{x,0,20}]

coefs=Rest[CoefficientList[series,x]]

Through[{Numerator,Denominator}[coefs]]

V7

SeriesCoefficient[ArcTan[x],{x,0,n}]

About 1

V5

<<DiscreteMath`RSolve`

SeriesTerm[ArcTan[1+x],{x,0,n}]

Table[(1/2) I If[n==0,-((I π)/2),If[-1+n≥0,((((-(1/2)-(I/2)))^n)/n)-((((-(1/2)+(I/2)))^n)/n),0]],{n,0,10}]

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

FullSimplify[Sum[(((2^(2n))(n!)^2)/((2n+1)!))((x^(2n+1))/(((1+x^2))^(n+1))),{n,0,∞}],x∈Reals]

FullSimplify[ArcSin[(x/(Sqrt[1+(x^2)]))],-1<x<1]

Plot[ArcSin[(x/(Sqrt[1+(x^2)]))]-ArcTan[x],{x,-5,5},PlotStyle->Red]

V7

Fibonacci Sums

(∑_(n=0)^∞)(((((-1))^n) (z^(2 n+1)))/(2 n+1))==(tan^(-1))(z)

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

(∑_(n=0)^∞)((((2^(2 n)) (((n!))^2)) (x^(2 n+1)))/((2 n+1)! ((((x^2)+1))^(n+1))))==(tan^(-1))(x)

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

FullSimplify[%]

FullSimplify[%-ArcTan[x]]

{(Sqrt[(1/(1+(x^2)))]) (Sqrt[1+(x^2)]) ArcSin[(x/(Sqrt[1+(x^2)]))],ArcTan[x]}/.x->1`50*^20 I//Chop

FullSimplify[(Sqrt[(1/(1+(x^2)))]) (Sqrt[1+(x^2)]) ArcSin[(x/(Sqrt[1+(x^2)]))],x>0]

FullSimplify[ArcSin[(x/(Sqrt[1+(x^2)]))]-ArcTan[x],x>0]

Table[ArcSin[(x/(Sqrt[1+(x^2)]))]-ArcTan[x]/.x->RandomReal[{0,10}],{1000}]//Chop//Union

Subtract@@@Table[{(Sqrt[(1/(1+(x^2)))]) (Sqrt[1+(x^2)]) ArcSin[(x/(Sqrt[1+(x^2)]))],ArcTan[x]}/.x->RandomComplex[],{100}]//Chop//Union

(∑_n^∞)(((i (2^(-n-1)) ((((-1-i))^n)-(((i-1))^n))) (x^n))/n)+(π/4)==(tan^(-1))(x)

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

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

FullSimplify[((I (2^(-1-n)) ((((-1-I))^n)-(((-1+I))^n)))/n),n∈Integers&&n>0]

FullSimplify[%]

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

FullSimplify[%]

Subtract@@@Table[{(1/4) (π+2 I Log[1+((1/2)-(I/2)) x]-2 I Log[1+((1/2)+(I/2)) x]),ArcTan[1+x]}/.x->RandomComplex[],{20}]//Chop//Union

With[{y=((x^2)/((x^2)+1))},(y/x) (∑_(n=0)^∞)(((2 n)!! (y^n))/((2 n+1)!!))]==(tan^(-1))(x)

With[{y=((x^2)/(1+(x^2)))},(y/x)Sum[(((2n)!!)/((2n+1)!!))y^n,{n,0,∞}]]

FullSimplify[%]

Table[(((2n)!!)/((2n+1)!!))y^n,{n,0,3}]

(∑_(n=0)^∞)(((((-1))^n) (t^(2 n+1)) (Subscript[F, 2 n+1]))/((5^n) (2 n+1)))==(tan^(-1))(x)

V6

s1=FullSimplify[With[{t=((2x)/(1+Sqrt[1+4x^2/5]))},Sum[(((((-1))^n)Fibonacci[2n+1](t^(2n+1)))/((5^n)(2n+1))),{n,0,∞}]
]]

FullSimplify[s1,x>0]

<<Simplify`

FullSimplify[TrigSimplify[1+s1-ArcTan[x]]]

Plot[s1-ArcTan[x],{x,-5,5},PlotStyle->Red]

V7

With[{t=((2x)/(1+Sqrt[1+4x^2/5]))},Sum[(((((-1))^n)Fibonacci[2n+1](t^(2n+1)))/((5^n)(2n+1))),{n,0,∞}]]

FullSimplify[%]

5 (∑_(n=0)^∞)(((((-1))^n) (Subscript[F, 2 n+1]^2))/((2 n+1) ((((Sqrt[(u^2)+1])+u))^(2 n+1))))==(tan^(-1))(x)

v4.2

s2=FullSimplify[With[{u=(5/(4x))(1+Sqrt[1+24x^2/25])},5Sum[(((((-1))^n)((Fibonacci[2n+1])^2))/((2n+1)(((u+Sqrt[u^2+1]))^(2n+1)))),{n,0,∞}]
]]

v5.0

s2=FullSimplify[With[{u=(5/(4x))(1+Sqrt[1+24x^2/25])},5Sum[(((((-1))^n)((Fibonacci[2n+1])^2))/((2n+1)(((u+Sqrt[u^2+1]))^(2n+1)))),{n,0,∞}]],x>0]

Plot[2 ArcTan[((4 x)/(5+(Sqrt[25+24 (x^2)])+(Sqrt[10]) (Sqrt[5+4 (x^2)+(Sqrt[25+24 (x^2)])])))]+ArcTan[((8 x)/((3+(Sqrt[5])) (5+(Sqrt[25+24 (x^2)])+(Sqrt[10]) (Sqrt[5+4 (x^2)+(Sqrt[25+24 (x^2)])]))))]+ArcTan[((2 (3+(Sqrt[5])) x)/(5+(Sqrt[25+24 (x^2)])+(Sqrt[10]) (Sqrt[5+4 (x^2)+(Sqrt[25+24 (x^2)])])))]-ArcTan[x],{x,0,5},PlotStyle->Red]

V7

With[{u=(5/(4x))(1+Sqrt[1+24x^2/25])},5Sum[(((((-1))^n)((Fibonacci[2n+1])^2))/((2n+1)(((u+Sqrt[u^2+1]))^(2n+1)))),{n,0,∞}]]

FullSimplify[%]

(∑_(n=0)^∞)(((((-1))^n) (5^(n+2)) (Subscript[F, 2 n+1]^3))/((2 n+1) ((((Sqrt[(v^2)+5])+v))^(2 n+1))))==(tan^(-1))(x)

Continued Fractions

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

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

x/(1+ContinuedFractionK[n^2x^2,2n+1,{n,20}])/.x->1.3

ArcTan[1.3]

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

{(x/(1+((x^2)/(3-(x^2))))),(x/(1+((x^2)/(3-(x^2)+((9(x^2))/(5-3(x^2)))))))}/.x->.9

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

With[{x=.9},x/(1+ContinuedFractionK[(2n-1)^2 x^2,(2n+1)-(2n-1)x^2,{n,40}])]

ArcTan[.9]

Recursion for Identies

a[x_,0]:=(1+x^2)^(-1/2)
b[x_,0]:=1
a[x_,i_]:=a[x,i]=1/2(a[x,i-1]+b[x,i-1])
b[x_,i_]:=b[x,i]=Sqrt[a[x,i]b[x,i-1]]
atan[x_,n_]:=x/Sqrt[1+x^2]/a[x,n]

atan[1.3,20]

ArcTan[1.3]