Inverse Tangent Integral

Author

Eric W. Weisstein
September 20, 2008

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

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

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

Definition

inverseTangentT[x_?NumericQ]:=(1/4) x LerchPhi[-(x^2),2,(1/2)]
inverseTangentT2[x_]:=(PolyLog[2,I x]-PolyLog[2,-I x])/(2I)

Plot

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

Real Axis

Plot[InverseTangentIntegralTi[x],{x,-10,10},
PlotStyle->Red,
AxesLabel->{x,HoldForm@InverseTangentIntegralTi[x]}]

Complex Plane

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

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

Limits

(lim_(x->∞)) (Subscript[Ti, 2])(x)

V5

InverseTangentIntegralTi[2,x]

Limit[InverseTangentT[2,x],x->#]&/@{-∞,∞}

Limit[(PolyLog[2,I x]-PolyLog[2,-I x])/(2I),x->#]&/@{-∞,∞}

V7

Limit[InverseTangentIntegralTi[x],x->#]&/@{-∞,∞}

Special Values

(Subscript[Ti, 2])( 0)==0

InverseTangentIntegralTi[0]

(Subscript[Ti, 2])( 1)==C

InverseTangentIntegralTi[1]

3 (Subscript[Ti, 2])( 1)-2 (Subscript[Ti, 2])((1/2))-(Subscript[Ti, 2])((1/3))-(1/2) (Subscript[Ti, 2])((3/4))==(1/2) π ln(2)

3InverseTangentIntegralTi[1]-2InverseTangentIntegralTi[1/2]-InverseTangentIntegralTi[1/3]-1/2InverseTangentIntegralTi[3/4]

FullSimplify[%]

N[{%,1/2π Log[2]}]

3 (Subscript[Ti, 2])(2-(Sqrt[3]))-2 (Subscript[Ti, 2])(1)==(1/4) π ln(2-(Sqrt[3]))

3 InverseTangentIntegralTi[2-(Sqrt[3])]-2 InverseTangentIntegralTi[1]

FullSimplify[%]

N[{%,π/4 Log[2-(Sqrt[3])]}]

(Subscript[Ti, 2])(tan((π/12)))-(2/3) (Subscript[Ti, 2])( tan((π/4)))==(1/12) π ln(tan((π/12)))

FullSimplify[InverseTangentIntegralTi[Tan[π/12]]-2/3InverseTangentIntegralTi[Tan[π/4]]]

N[{%,1/12π Log[Tan[π/12]]}]

3 (Subscript[Ti, 2])((Sqrt[3])+2)-2 (Subscript[Ti, 2])( 1)==(5/4) π ln((Sqrt[3])+2)

FullSimplify[3InverseTangentIntegralTi[2+(Sqrt[3])]-2InverseTangentIntegralTi[1]]

N[{%,5/4π Log[2+(Sqrt[3])]}]

5.1+0.I

-(1/3) 2 (Subscript[Ti, 2])(tan((π/8)))-(Subscript[Ti, 2])(tan((π/24)))+(Subscript[Ti, 2])(tan(((5 π)/24)))==(1/6) π ln(((tan(((5 π)/24)))/(tan((π/8)))))

FullSimplify[-InverseTangentIntegralTi[Tan[π/24]]+InverseTangentIntegralTi[Tan[5π/24]]-2/3InverseTangentIntegralTi[Tan[π/8]]
]

N[{%,π/6Log[Tan[5π/24]/Tan[π/8]]}]//Chop

-(Subscript[Ti, 2])((((Sqrt[3])-(Sqrt[2]))/((Sqrt[2])-1)))+(2/3) (Subscript[Ti, 2])((Sqrt[2])-1)+(Subscript[Ti, 2])((((Sqrt[3])-(Sqrt[2]))/((Sqrt[2])+1)))==(1/6) π ln((((Sqrt[2])-1)/(((Sqrt[3])-(Sqrt[2])) ((Sqrt[2])+1))))

FullSimplify[InverseTangentIntegralTi[(((Sqrt[3])-(Sqrt[2]))/((Sqrt[2])+1))]-InverseTangentIntegralTi[(((Sqrt[3])-(Sqrt[2]))/((Sqrt[2])-1))]+(2/3)InverseTangentIntegralTi[(Sqrt[2])-1]]

N[{%,(π/6)Log[(((Sqrt[2])-1)/(((Sqrt[3])-(Sqrt[2]))((Sqrt[2])+1)))]}]//Chop

Sums

(∑_k^∞)(((((-1))^(k-1)) (x^(2 k-1)))/(((2 k-1))^2))==(1/4) x Φ(-(x^2),2,(1/2))

V6

t1=(∑_k^∞)(((((-1))^(k-1)) (x^(2 k-1)))/(((2 k-1))^2))

V7

(∑_k^∞)(((((-1))^(k-1)) (x^(2 k-1)))/(((2 k-1))^2))

FunctionExpand[%]

(∑_(n=1)^∞)((sin((4 n-2) x))/(((2 n-1))^2))==(1/2) i (e^(-2 i x)) ((Subscript[, 3])(Subscript[F, 2])((1/2),(1/2),1;(3/2),(3/2);(e^(-4 i x)))-(e^(4 i x)) (Subscript[, 3])(Subscript[F, 2])((1/2),(1/2),1;(3/2),(3/2);(e^(4 i x))))==(Subscript[Ti, 2])(tan(x))-x ln(tan(x))

(∑_(n=1)^∞)((Sin[(4 n-2) x])/(((2 n-1))^2))

FullSimplify[%]

InverseTangentIntegralTi[Tan[x]]- x Log[Tan[x]]

expr=FullSimplify[Subtract@@{%389,InverseTangentIntegralTi[Tan[x]]- x Log[Tan[x]]}]

FunctionExpand[expr]//FullSimplify

Table[expr/.x->RandomComplex[],{100}]//Chop//Union

Identities

(Subscript[Ti, 2]) (x tan)-ln x (x tan)==(1/2) i (e^(-2 i x)) (HypergeometricPFQ((Subscript[, 3])(Subscript[F, 2])((1/2),(1/2),1;(3/2),(3/2);(e^(-4 i x))))-(e^(4 i x)) HypergeometricPFQ((Subscript[, 3])(Subscript[F, 2])((1/2),(1/2),1;(3/2),(3/2);(e^(4 i x)))))==-ln x (x tan)-(1/2) i ((Subscript[PolyLog, 2]) (i x tan)-(Subscript[PolyLog, 2]) (-i x tan))

(Subscript[Ti, 2])(x)==(1/4) x Φ(-(x^2),2,(1/2))

t2=(PolyLog[2,I x]-PolyLog[2,-I x])/(2I)

FullSimplify[(1/4) (x LerchPhi[-(x^2),2,(1/2)]+2 I (-PolyLog[2,-I x]+PolyLog[2,I x]))]

Table[t1-t2 /.x->Random[Real,{0,10}],{10}]//Chop

Table[(1/4) (x LerchPhi[-(x^2),2,(1/2)]+2 I (-PolyLog[2,-I x]+PolyLog[2,I x]))/.x->RandomComplex[],{100}]//Chop//Union

(Subscript[Ti, 2])(x)-(Subscript[Ti, 2])( (1/x))==(1/2) π sgn(x) ln(|x|)

HoldForm[InverseTangentIntegralTi[x]]//TraditionalForm

FullSimplify[InverseTangentIntegralTi[x]-InverseTangentIntegralTi[(1/x)],x<0]

Refine[(1/2)π Sign[x]Log[Abs[x]],x<0]

FullSimplify[InverseTangentIntegralTi[x]-InverseTangentIntegralTi[(1/x)],x>0]

FullSimplify[(1/2)π Sign[x]Log[Abs[x]],x>0]

Subtract@@@Table[{InverseTangentIntegralTi[x]-InverseTangentIntegralTi[(1/x)],(1/2)π Sign[x]Log[Abs[x]]}/.x->RandomReal[{-2,2}],{100}]//Chop//Union

(1/2) (Subscript[Ti, 2])( ((2 x)/(1-(x^2))))==??InverseTangentIntegralTi(-x,1)-InverseTangentIntegralTi(x,1)+(Subscript[Ti, 2])( x)

InverseTangentIntegralTi[x_,1]:=(1/96) (-48 Catalan+I (π^2)+24 (-4 I ((ArcTan[x])^2)+ArcTan[x] (5 I π-4 Log[((2 I)/(-I+x))]+4 Log[(1+I)+(2/(I+x))])+π (2 Log[-((2 I)/(-I+x))]+Log[((1/8)+(I/8)) (-I+x) (1+x) Csc[(π/4)+ArcTan[x]]]))-48 I PolyLog[2,((I+x)/(-I+x))]-48 I PolyLog[2,-I-(2/(I+x))])

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

FullSimplify[Subtract@@%]

Table[(1/4) I (2 ArcTan[x] (5 π+2 I Log[((2 I)/(-I+x))]+2 I Log[-((2 I)/(I+x))]-2 I Log[(1+I)-(2/(-I+x))]-2 I Log[(1+I)+(2/(I+x))])-I π (2 Log[-((2 I)/(-I+x))]-2 Log[((2 I)/(I+x))]-Log[-(1/4) (((-1))^(1/4)) (I+x) (Sqrt[1+(x^2)])]+Log[((1/8)+(I/8)) (-I+x) (1+x) Csc[(π/4)+ArcTan[x]]])-2 PolyLog[2,-I x]+2 PolyLog[2,I x]+2 PolyLog[2,((-I+x)/(I+x))]-2 PolyLog[2,((I+x)/(-I+x))]-PolyLog[2,-((2 I x)/(-1+(x^2)))]+PolyLog[2,((2 I x)/(-1+(x^2)))]+2 PolyLog[2,-I+(2/(-I+x))]-2 PolyLog[2,-I-(2/(I+x))])/.x->RandomReal[],{10}]//Chop

(1/3) (Subscript[Ti, 2])(((3 x-(x^2))/(1-3 (x^2))))==??(1/6) π ln((((x+(Sqrt[3])) ((Sqrt[3]) x+1))/((1-(Sqrt[3]) x) ((Sqrt[3])-x))))-(Subscript[Ti, 2])((((Sqrt[3]) x+1)/((Sqrt[3])-x)))+(Subscript[Ti, 2])(x)+(Subscript[Ti, 2])(((1-(Sqrt[3]) x)/(x+(Sqrt[3]))))

{(1/3)InverseTangentIntegralTi[((3x-(x^2))/(1-3(x^2)))],InverseTangentIntegralTi[x]+InverseTangentIntegralTi[((1-x(Sqrt[3]))/((Sqrt[3])+x))]-InverseTangentIntegralTi[((1+x(Sqrt[3]))/((Sqrt[3])-x))]+(π/6)Log[((((Sqrt[3])+x)(1+x(Sqrt[3])))/((1-x(Sqrt[3]))((Sqrt[3])-x)))]}

FullSimplify[Subtract@@%]

(1/6) (-π Log[1+((8 x)/((Sqrt[3])+x (-4+(Sqrt[3]) x)))]-3 I PolyLog[2,-I x]+3 I PolyLog[2,I x]+3 I PolyLog[2,-((I (-1+(Sqrt[3]) x))/((Sqrt[3])+x))]-3 I PolyLog[2,((I (-1+(Sqrt[3]) x))/((Sqrt[3])+x))]+3 I PolyLog[2,-((I (1+(Sqrt[3]) x))/((Sqrt[3])-x))]-3 I PolyLog[2,((I (1+(Sqrt[3]) x))/((Sqrt[3])-x))]+I PolyLog[2,-((I (-3+x) x)/(-1+3 (x^2)))]-I PolyLog[2,((I (-3+x) x)/(-1+3 (x^2)))])/.x->RandomReal[]

Integral

(∫_0^x)(((tan^(-1))(t))/t)dt

Integrate[ArcTan[xx]/xx,{xx,0,x},Assumptions->x>0]

(∫_0^x)(((tan^(-1))(t))/(a+t))dt

V5

FullSimplify[Integrate[ArcTan[xx]/(a+xx),{xx,0,x},Assumptions->x>0]]

V7

FullSimplify[Integrate[ArcTan[xx]/(a+xx),{xx,0,x},Assumptions->x>0]]

(∫_0^x)(((tan^(-1))(t))/(1+t))dt

FullSimplify[FullSimplify[Integrate[ArcTan[xx]/(1+xx),{xx,0,x},Assumptions->x>0]],x>0]

Derivative

FullSimplify[D[InverseTangentIntegralTi[x],x]]

Integrate[((ArcTan[x])/x),x]

FullSimplify[%-InverseTangentIntegralTi[x]]