ContinuedDiLog

({{, ContinuedDiLog[{(Subscript[x, 1]),(Subscript[α, 1])},{(Subscript[x, 2]),(Subscript[α, 2])}] 
gives the two variable Beenakker-Denner continued dilogarithm function (Subscript[ℒ𝒾, 2])((Subscript[x, 1])+i (Subscript[α, 1])ε, (Subscript[x, 2])+i (Subscript[α, 2])ε).}})

Details

Mathematical function, suitable for both symbolic and numerical manipulation.

The function (Subscript[ℒ𝒾, 2])((Subscript[x, 1]), (Subscript[x, 2])) has the contour integral representation
	(Subscript[ℒ𝒾, 2])((Subscript[x, 1]), (Subscript[x, 2]))=(∫_( C))((Ln(t))/(1-t)) dt,
where C is the contour that starts at t=1 on the principal sheet of the complex logarithm Ln(t) and ends at t=(Subscript[x, 1])(Subscript[x, 2]) on the Riemann sheet of the logarithm determined by the accumulated phase of the product (Subscript[x, 1])(Subscript[x, 2]).

If -π <arg((Subscript[x, 1]))+arg((Subscript[x, 2]))<π, (Subscript[ℒ𝒾, 2])((Subscript[x, 1]), (Subscript[x, 2]))=(Subscript[Li, 2])(1-(Subscript[x, 1])(Subscript[x, 2])).

ContinuedDiLog[{(Subscript[x, 1]),(Subscript[α, 1])},{(Subscript[x, 2]),(Subscript[α, 2])}] has a branch cut discontinuity in each complex (Subscript[x, i]) plane running from 0 to -∞.

For real negative product (Subscript[x, 1])(Subscript[x, 2])<0, the sign of (Subscript[x, 1])(Subscript[α, 2])+(Subscript[x, 2])(Subscript[α, 1]) determines on which side of the branch cut ContinuedDiLog evaluates.

Examples

Basic Examples  (3)

Evaluate numerically:

ContinuedDiLog[{3.2,1.0},{1.1,1.0}]

Evaluate symbolically:

ContinuedDiLog[{3+2I,1},{1/10,1}]

ContinuedDiLog[{3+2I,1},{0,1}]

Compare the principal value of the dilogarithm function with the analytically continued version:

x1=-3.0+1.0I;
x2=1.0-0.5I;

PolyLog[2,1-x1 x2]
ContinuedDiLog[{x1,1},{x2,1}]

x1=-3.0+1.0 I;
x2=-2.0+1.0I;

PolyLog[2,1-x1 x2]
ContinuedDiLog[{x1,1},{x2,1}]

Properties & Relations  (4)

Evaluate to high precision:

ContinuedDiLog[{3.2`50,1.0},{1.1`50,1.0}]

ContinuedDiLog is needed to analytically represent the Passarino-Veltman D-functions:

ScalarD0[0,0,0,(m^2),s,t,m,0,m,0]//D0Expand

ScalarD0IR12[(M^2),(M^2),s,t,m,m]//D0Expand

The derivative:

D[ContinuedDiLog[{x,xInf},{y,yInf}],x]

PolyLog has a branch cut discontinuity when the phase of the product (Subscript[x, 1])(Subscript[x, 2]) crosses π, whereas ContinuedDiLog is continuous over the interval -π to +π for the phase of each variable (Subscript[x, 1]) and (Subscript[x, 2]) separately:

Grid[{{"PolyLog","ContinuedDiLog"},{Plot[Im[PolyLog[2,1-(E^(I π/2))(E^(I θ))]],{θ,-π,π},PlotRange->{-5,5},PlotStyle->Red],
Plot[Im[ContinuedDiLog[{(E^(I π/2)),1},{(E^(I θ)),1}]],{θ,-π,π},PlotRange->{-5,5},PlotStyle->Blue]}}]

See Also

Related Guides