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HarmonicPolyLog

HarmonicPolyLog[{a₁,a₂,…,a_k},x]

gives the harmonic polylogarithm $TemplateBox[{TemplateBox[{{{a, _, 1}, ,, {a, _, 2}, ,, ..., ,, {a, _, k}}}, ImplicitList], x}, HarmonicPolyLog]$ .

HarmonicPolyLog[{a₁,a₂,…,a_k},x, p]

gives the harmonic polylogarithm with base point p.

HarmonicPolyLog[{a₁,a₂,…,a_k},x,{p₁,p₂,…,p_k}]

gives the harmonic polylogarithm with a sequence of base points (p₁, p₂, …).

Details

HarmonicPolyLog gives depth- Chen iterated integrals built from the kernels , and , unifying logs and classical polylogarithms.
Let , , and . Then HarmonicPolyLog with letters is defined, when it converges, as the following iterated integral:
$TemplateBox[{TemplateBox[{{{a, _, 1}, ,, {a, _, 2}, ,, ..., ,, , {a, _, k}}}, ImplicitList], x}, HarmonicPolyLog]$

$TemplateBox[{TemplateBox[{{{a, _, 1}, ,, {a, _, 2}, ,, ..., ,, , {a, _, k}}}, ImplicitList], {x, ;, , p}}, HarmonicPolyLog]$

$TemplateBox[{TemplateBox[{{{a, _, 1}, ,, {a, _, 2}, ,, ..., ,, , {a, _, k}}}, ImplicitList], {x, ;, , {{, {{p, _, 1}, ,, , {p, _, 2}, ,, , ..., ,, , {p, _, k}}, }}}}, HarmonicPolyLog]$
For example, is .
The following abbreviated notation is used in the first argument: block is replaced by and block is replaced by , where denotes a sequence of zeros. Thus, is same as .
The first argument of HarmonicPolyLog[{a₁, a₁, … a_k}, x] is the word, and are the letters (or weights). x is the endpoint.
The defining iterated integral of $TemplateBox[{TemplateBox[{{{a, _, 1}, ,, {a, _, 2}, ,, ..., ,, {a, _, k}}}, ImplicitList], x}, HarmonicPolyLog]$ (assuming are any of , and ) diverges in the following three conditions:
Lower-endpoint collision

Upper-endpoint collision

Letter crossing (pole on the path) at least one of lies on the open segment
The following measures are taken in the divergent cases:

	Lower-endpoint collision	returns a regularized value using shuffle relations
	Upper-endpoint collision	returns ComplexInfinity
	Letter crossing (pole on the path)	returns values obtained by analytic continuation, using infinitesimal circular detours around the poles

HarmonicPolyLog is a mathematical function, suitable for both symbolic and numerical manipulation.
The function supports numerical evaluation of the following types:
machine precision N Yes

arbitrary precision N Yes

worst-case intervals CenteredInterval No

average-case intervals Around No
The following are reductions of HarmonicPolyLog for the basic letters:
HarmonicPolyLog[{}, x]=1

HarmonicPolyLog[{0}, x]=Log[x]

HarmonicPolyLog[{1}, x]=-Log[1-x]

HarmonicPolyLog[{-1}, x]=Log[1+x]
Products at the same endpoint satisfy the shuffle algebra: $TemplateBox[{u, x}, HarmonicPolyLog]*TemplateBox[{v, x}, HarmonicPolyLog]= sum_(w in u ⧢ v)TemplateBox[{w, x}, HarmonicPolyLog]$ , where denotes all interleavings preserving the internal orders of and .
While MultiplePolyLog is the generalization of the series definition of PolyLog, HarmonicPolyLog represents a generalization of the integral definition of PolyLog.
Sum, GeneratingFunction, ExponentialGeneratingFunction, ZTransform, DirichletTransform, Integrate, DSolve and AsymptoticDSolveValue may return results in terms of HarmonicPolyLog.

Examples

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Basic Examples (5)

Evaluate numerically:

Wolfram Language code: HarmonicPolyLog[{-1, 0, 1}, 1]

The defining integral:

Wolfram Language code: Integrate[(1/x1 x2 (1 + x3) x4(1 - x5)), {x1, 0, (1/2)}, {x2, 0, x1}, {x3, 0, x2}, {x4, 0, x3}, {x5, 0, x4}]

Plot over a subset of the reals:

Wolfram Language code: Plot[HarmonicPolyLog[{-1, 0, 0, 1}, x], {x, -1 / 2, 1 / 2}]

Plot over a subset of complexes:

Wolfram Language code: ComplexPlot3D[HarmonicPolyLog[{-1, 1, -1}, x], {x, -1 / 2 - I / 2, 1 / 2 + I / 2}, PlotPoints -> 10]

Series expansion at the origin:

Wolfram Language code: Series[HarmonicPolyLog[{-1, 1}, x], {x, 0, 10}]

Scope (17)

Numerical Evaluation (2)

Evaluate numerically:

Wolfram Language code: HarmonicPolyLog[{-1, 0, 1}, 1 / 2.]

Evaluate to high precision:

Wolfram Language code: N[HarmonicPolyLog[{-1, 0, 1}, 1 / 2], 20]

Specific Values (4)

A single zero letter reproduces the principal logarithm:

Wolfram Language code: HarmonicPolyLog[{0}, 10]

Word composed of three letters:

Wolfram Language code: HarmonicPolyLog[{-1, -1, 1}, 1 / 2]

Word composed of five letters:

Wolfram Language code: HarmonicPolyLog[{0, -1, 1, 1, 1}, 1]

Evaluate at a complex endpoint:

Wolfram Language code: HarmonicPolyLog[{-1, -1, 1}, 2 - 4I]

Visualization (6)

Plot the real and imaginary parts of a depth-2 HarmonicPolyLog along the real axis:

Wolfram Language code: ReImPlot[HarmonicPolyLog[{-1, 1}, x], {x, -1, 1}]

Evaluate the same HarmonicPolyLog along the slanted segment :

Wolfram Language code:

pathPlot = Graphics[{Thick, Arrowheads[.15], Arrow[{{0, 0}, {1, 1}}]}, ...];
ReImPlot[GeneralizedPolyLog[{-1, 1}, (1 + I) x], {x, 0, 1}, ...]

Trace the HarmonicPolyLog along the circle :

Wolfram Language code:

pathCirclePlot = Graphics[{Circle[{0, 0}, 1 / 2, {-Pi / 2, 3 Pi / 2}] }, ...];
ParametricPlot[ReIm@HarmonicPolyLog[{-1, 1}, (1 / 2) Exp[I t]], {t, 0, 2 Pi}, ...]

See the evolution with four different radii:

Wolfram Language code:

With[{rs = {.3, .45, .6, .75}}, ParametricPlot[Evaluate@Table[ReIm@HarmonicPolyLog[{-1, 1}, r Exp[I t]], {r, rs}], {t, 0, 2 Pi}, ...]]

Check the effect of incrementally adding new symmetric letters:

Wolfram Language code:

GraphicsRow[{ParametricPlot[ReIm[HarmonicPolyLog[{1}, (1 / 2)Exp[I t]]], {t, 0, 2Pi}, Rule[...]], ParametricPlot[ReIm[HarmonicPolyLog[{-1, 1}, (1 / 2)Exp[I t]]], {t, 0, 2Pi}, Rule[...]], ParametricPlot[ReIm[HarmonicPolyLog[{-1, -1, 1, 1}, (1 / 2)Exp[I t]]], {t, 0, 2Pi}, Rule[...]], 
	ParametricPlot[ReIm[HarmonicPolyLog[{-1, -1, -1, 1, 1, 1}, (1 / 2)Exp[I t]]], {t, 0, 2Pi}, Rule[...]]}
	]

Zero contours of real and imaginary parts of HarmonicPolyLog[{1, -1},z]:

Wolfram Language code:

Clear[g];g[z_] := HarmonicPolyLog[{1, -1}, z];
ContourPlot[{Re[g[x + I y]] == 0, Im[g[x + I y]] == 0}, {x, -1 / 2, 1 / 2}, {y, -1 / 2, 1 / 2}, ...]

Differentiation (4)

Take derivative w.r.t. the endpoint:

Wolfram Language code: D[HarmonicPolyLog[{1, -1, 0, 0, -1, 0, 0, 1}, x], x]

Take higher-order derivative:

Wolfram Language code: D[HarmonicPolyLog[{1, -1, 0, 0, -1, 0, 0, 1}, x ^ 2], {x, 3}]

Endpoint is a function of :

Wolfram Language code: D[HarmonicPolyLog[{1, -1, 0, 0, -1, 0, 0, 1}, x ^ 3], x]

Take derivative of a HarmonicPolyLog having a non-zero basepoint w.r.t. the endpoint:

Wolfram Language code: D[HarmonicPolyLog[{2, 3, 4}, x, I], x]

Integration (1)

Indefinite integration involving HarmonicPolyLog:

Wolfram Language code: Integrate[HarmonicPolyLog[{2, 3}, x], x]

Definite integration involving HarmonicPolyLog:

Wolfram Language code: Integrate[HarmonicPolyLog[{2, 3}, x], {x, 0, 1}]

Generalizations & Extensions (7)

Regularization (3)

Upper-endpoint collision—HarmonicPolyLog diverges when the endpoint coincides with the first letter:

Wolfram Language code: HarmonicPolyLog[{1, 2}, 1]

Lower-endpoint collision—trailing zeros in the word cause divergence but can be regularized:

Wolfram Language code: HarmonicPolyLog[{1, 2, 0}, 1 / 2]

Wolfram Language code: HarmonicPolyLog[{1, 0, 0}, 1 / 2]

With this regularization, one can continue to use the iterated integral definition with Log.

A regularized HarmonicPolyLog:

Wolfram Language code: HarmonicPolyLog[{-1, 0, 0}, x]

Evaluate at :

Wolfram Language code: % /. x -> 1.I

This is the same as:

Wolfram Language code: NIntegrate[(1 / (1 + x))Log[x] ^ 2 / Factorial[2], {x, 0, I}]

Analytic continuation (2)

The defining integral for HarmonicPolyLog[{1,-1}, 2] diverges, since the pole at lies on the open segment :

Wolfram Language code: Integrate[(1/1 - x1)(1/x2 + 1), {x1, 0, 2}, {x2, 0, x1}]

A finite value is obtained by integrating along an infinitesimal circular detour around the pole:

Wolfram Language code: HarmonicPolyLog[{1, -1}, 2]//N

Visualize the branch-cut discontinuity of HarmonicPolyLog[{1,-1}, x]:

Wolfram Language code: ReImPlot[HarmonicPolyLog[{1, -1}, x], {x, 0, 2}]

Nonzero Base Point (2)

A HarmonicPolyLog with base point I:

Wolfram Language code: HarmonicPolyLog[{1, -1}, 1 / 2, I]

Obtain the same from its integral representation:

Wolfram Language code: % == Integrate[(1/(1 - x1)(1 + x2)), {x1, I, 1 / 2}, {x2, I, x1}]

A HarmonicPolyLog with a sequence of base points {I, 2 I}:

Wolfram Language code: HarmonicPolyLog[{1, -1}, 1 / 2, {I, 2I}]

Obtain the same from its integral representation:

Wolfram Language code: % == Integrate[(1/(1 - x1)(1 + x2)), {x1, I, 1 / 2}, {x2, 2I, x1}]

Applications (11)

Euler Sum (3)

Compute an Euler sum involving HarmonicNumber in terms of HarmonicPolyLog:

Wolfram Language code: Underoverscript[∑, n = 1, ∞](HarmonicNumber[4 n, 2]/n^4)

Compute an Euler sum involving AlternatingHarmonicNumber in terms of HarmonicPolyLog:

Wolfram Language code: Underoverscript[∑, n = 1, ∞](AlternatingHarmonicNumber[n]^3/n^3)

An Euler sum involving a product of a HarmonicNumber and an AlternatingHarmonicNumber:

Wolfram Language code: Underoverscript[∑, n = 1, ∞](HarmonicNumber[n] AlternatingHarmonicNumber[n]/2^n n^2)

Generating Function (2)

GeneratingFunction of AlternatingHarmonicNumber is expressed in terms of HarmonicPolyLog:

Wolfram Language code: GeneratingFunction[AlternatingHarmonicNumber[n, 2] ^ 2, n, x]

ExponentialGeneratingFunction of products of StirlingS1 and AlternatingHarmonicNumber is expressed in terms of HarmonicPolyLog:

Wolfram Language code: ExponentialGeneratingFunction[StirlingS1[n, 3]AlternatingHarmonicNumber[n, 2], n, x]

Discrete Transforms (2)

ZTransform of AlternatingHarmonicNumber is often expressed in terms of HarmonicPolyLog:

Wolfram Language code: ZTransform[AlternatingHarmonicNumber[n, 2] ^ 2, n, z]

DirichletTransform of AlternatingHarmonicNumber is often expressed in terms of HarmonicPolyLog:

Wolfram Language code: DirichletTransform[AlternatingHarmonicNumber[n, 2] ^ 3, n, 2]

Iterated Integrals (2)

A rational iterated integral is computed in terms of HarmonicPolyLog and GeneralizedPolyLog:

Wolfram Language code: Integrate[(1/x1 (1 + x2) (1 + x3)(1 - x4)(10 - x4)), {x1, 0, 1 / 2}, {x2, 0, x1}, {x3, 0, x2}, {x4, 0, x3}]

A more complex integral:

Wolfram Language code:

Short[Integrate[(1/x1 x2(1 + x2)^2 (1 - x3) x4(1 + x5)^2(1 - x6)x7(1 - x8)), {x1, 0, 1}, {x2, 0, x1}, {x3, 0, x2}, {x4, 0, x3}, {x5, 0, x4}, {x6, 0, x5}, {x7, 0, x6}, {x8, 0, x7}], 4]

Rational ODE System (2)

Solution of a Fuchsian system of ordinary differential equations is expressed in terms of HarmonicPolyLog:

Wolfram Language code:

DSolveValue[{Derivative[1][y1][x] == ((2/-1 + x) + (1/1 + x)) y2[x] + ((3/-1 + x) + (2/1 + x)) y3[x] + ((4/-1 + x) + (3/1 + x)) y4[x], Derivative[1][y2][x] == ((3/-1 + x) + (1/1 + x)) y3[x] + ((4/-1 + x) + (2/1 + x)) y4[x], Derivative[1][y3][x] == ((2/-1 + x) + (1/1 + x)) y4[x], Derivative[1][y4][x] == 0, y1[0] == 1, y2[0] == 1, y3[0] == 1, y4[0] == 1}, {y1[x], y2[x], y3[x], y4[x]}, x]

Solution of a perturbative rational ODE system is expressed in terms of HarmonicPolyLog:

Wolfram Language code:

AsymptoticDSolveValue[{Derivative[1][y1][x] == ϵ ((3 y1[x]/1 + x) + ((1/-1 + x) + (2/1 + x)) y2[x] + ((2/(-1 + x)^2) + (3/1 + x)) y3[x]), Derivative[1][y2][x] == ϵ (-(4 y1[x]/-1 + x) + ((3/-1 + x) + (2/1 + x)) y3[x]), Derivative[1][y3][x] == 0, y1[0] == 1, y2[0] == 2, y3[0] == 3}, {y1[x], y2[x], y3[x]}, x, {ϵ, 0, 3}]

Properties & Relations (10)

An empty word produces unity:

Wolfram Language code: HarmonicPolyLog[{}, x]

Depth-1 HarmonicPolyLog with letter reproduces the natural logarithm:

Wolfram Language code: HarmonicPolyLog[{0}, x]

Words composed of repeated zeros give rise to powers of Log:

Wolfram Language code: HarmonicPolyLog[{0, 0, 0}, x]

A block of zeros followed by a collapses to a classical polylogarithm:

Wolfram Language code: HarmonicPolyLog[{0, 0, 0, 1}, x]

A block of zeros followed by a block of identical s collapses to a Nielsen polylogarithm:

Wolfram Language code: HarmonicPolyLog[{0, 0, 0, 0, 1, 1}, x]

Words over evaluated at give MultipleZeta:

Wolfram Language code: HarmonicPolyLog[{0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1}, 1]

GeneralizedPolyLog with words over gives HarmonicPolyLog:

Wolfram Language code: GeneralizedPolyLog[{-1, 0, -1, -1, 0, 0, 1, 1, 0, 1, 1}, 1 / 2]

Express Log of polynomial with roots , and as a rational linear combination of HarmonicPolyLog instances (plus a piecewise constant):

Wolfram Language code: logPolynomial = Log[x - x^2 - x^3 + x^4];

Wolfram Language code:

logPolyAsHPL = ResourceFunction["LogAffineExpand"][logPolynomial, x] /. {
	Log[1 + x] :> Inactive[HarmonicPolyLog][{-1}, x], Log[1 - x] :> -Inactive[HarmonicPolyLog][{1}, x], Log[x] :> Inactive[HarmonicPolyLog][{0}, x]
	}

Wolfram Language code: PossibleZeroQ[Chop[N@Activate[(logPolynomial - logPolyAsHPL) /. x -> 2]]]

Similarly for Log of a rational function:

Wolfram Language code: logRational = Log[(1 - x) / (x - x^2 - x^3 + x^4)];

Wolfram Language code:

logRatAsHPL = ResourceFunction["LogAffineExpand"][logRational, x] /. {
	Log[1 + x] :> Inactive[HarmonicPolyLog][{-1}, x], Log[1 - x] :> -Inactive[HarmonicPolyLog][{1}, x], Log[x] :> Inactive[HarmonicPolyLog][{0}, x]}

Wolfram Language code: PossibleZeroQ[Chop[N@Activate[(logRational - logRatAsHPL) /. x -> 10]]]

HarmonicPolyLog appears as the limit of MultipleHarmonicNumber as the index goes to :

Wolfram Language code: MultipleHarmonicNumber[Infinity, {3, 2, 1}, {1, -1, 1}]

Possible Issues (1)

The letters of a HarmonicPolyLog must be integers:

Wolfram Language code: HarmonicPolyLog[{I, 2I}, x]

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HarmonicPolyLog

Details

Examples

Basic Examples (5)

Scope (17)

Numerical Evaluation (2)

Specific Values (4)

Visualization (6)

Differentiation (4)

Integration (1)

Generalizations & Extensions (7)

Regularization (3)

Analytic continuation (2)

Nonzero Base Point (2)

Applications (11)

Euler Sum (3)

Generating Function (2)

Discrete Transforms (2)

Iterated Integrals (2)

Rational ODE System (2)

Properties & Relations (10)

Possible Issues (1)

Text

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APA

BibTeX

BibLaTeX

machine precision	N	Yes
arbitrary precision	N	Yes
worst-case intervals	CenteredInterval	No
average-case intervals	Around	No

	HarmonicPolyLog[{}, x]=1
	HarmonicPolyLog[{0}, x]=Log[x]
	HarmonicPolyLog[{1}, x]=-Log[1-x]
	HarmonicPolyLog[{-1}, x]=Log[1+x]

HarmonicPolyLog

Details

Examples

Basic Examples (5)

Scope (17)

Numerical Evaluation (2)

Specific Values (4)

Visualization (6)

Differentiation (4)

Integration (1)

Generalizations & Extensions (7)

Regularization (3)

Analytic continuation (2)

Nonzero Base Point (2)

Applications (11)

Euler Sum (3)

Generating Function (2)

Discrete Transforms (2)

Iterated Integrals (2)

Rational ODE System (2)

Properties & Relations (10)

Possible Issues (1)

See Also

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX