{"history":[],"article":{"id":"51835dd7-3af9-4382-88b4-27a6f14d06f7","title":"Polylogarithm","slug":"polylogarithm","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **polylogarithm** (also known as **Jonquière's function**, for Alfred Jonquière) is a [special function](https://bliptext.com/articles/special-function)\nLis(z)\nof order s and argument z. Only for special values of s does the polylogarithm reduce to an [elementary function](https://bliptext.com/articles/elementary-function) such as the [natural logarithm](https://bliptext.com/articles/natural-logarithm) or a [rational function](https://bliptext.com/articles/rational-function).\nIn [quantum statistics](https://bliptext.com/articles/quantum-statistics), the polylogarithm function appears as the closed form of [integrals](https://bliptext.com/articles/integral) of the [Fermi–Dirac distribution](https://bliptext.com/articles/fermi-dirac-distribution) and the [Bose–Einstein distribution](https://bliptext.com/articles/bose-einstein-distribution), and is also known as the **Fermi–Dirac integral** or the **Bose–Einstein integral**.\nIn [quantum electrodynamics](https://bliptext.com/articles/quantum-electrodynamics), polylogarithms of positive [integer](https://bliptext.com/articles/integer) order arise in the calculation of processes represented by higher-order [Feynman diagrams](https://bliptext.com/articles/feynman-diagram).\nThe polylogarithm [function](https://bliptext.com/articles/function-mathematics) is equivalent to the [Hurwitz zeta function](https://bliptext.com/articles/hurwitz-zeta-[function](https://bliptext.com/articles/function-mathematics)) — either [function](https://bliptext.com/articles/function-mathematics) can be expressed in terms of the other — and both [function](https://bliptext.com/articles/function-mathematics)s are special cases of the [Lerch transcendent](https://bliptext.com/articles/lerch-transcendent).\nPolylogarithms should not be confused with [polylogarithmic functions](https://bliptext.com/articles/polylogarithmic-function), nor with the [offset logarithmic integral](https://bliptext.com/articles/offset-logarithmic-integral)\nLi(z)\n, which has the same notation without the subscript.\nThe polylogarithm function is defined by a [power series](https://bliptext.com/articles/power-series) in z, which is also a [Dirichlet series](https://bliptext.com/articles/dirichlet-series) in s:\nThis definition is valid for arbitrary [complex](https://bliptext.com/articles/complex-numbers) order s and for all [complex](https://bliptext.com/articles/complex-numbers) arguments z with\nby the process of [analytic continuation](https://bliptext.com/articles/analytic-continuation).\n(Here the denominator ks is understood as\nexp(s ln k)\n). The special case\ns = 1\ninvolves the ordinary [natural logarithm](https://bliptext.com/articles/natural-logarithm),\nLi1(z) = −ln(1−z)\n, while the special cases\ns = 2\nand\ns = 3\nare called the [dilogarithm](https://bliptext.com/articles/dilogarithm) (also referred to as Spence's function) and trilogarithm respectively.\nThe name of the function comes from the fact that it may also be defined as the repeated [integral](https://bliptext.com/articles/indefinite-integral) of itself:\nthus the dilogarithm is an integral of a function involving the logarithm, and so on.\nFor nonpositive integer orders s, the polylogarithm is a [rational function](https://bliptext.com/articles/rational-function).\n\n# Properties\nIn the case where the order s is an integer, it will be represented by s=n (or s=-n when negative).\nIt is often convenient to define \\mu=\\ln(z) where \\ln(z) is the [principal branch](https://bliptext.com/articles/principal-branch) of the [complex logarithm](https://bliptext.com/articles/complex-logarithm) so that Also, all exponentiation will be assumed to be single-valued:\nDepending on the order s, the polylogarithm may be multi-valued.\nThe principal branch of is taken to be given for | z | < 1 by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from z = 1 to \\infty such that the axis is placed on the lower half plane of z. In terms of \\mu, this amounts to.\nThe discontinuity of the polylogarithm in dependence on \\mu can sometimes be confusing.\nFor real argument z, the polylogarithm of real order s is real if z < 1, and its imaginary part for z \\ge 1 is :\nGoing across the cut, if ε is an infinitesimally small positive real number, then:\nBoth can be concluded from the series expansion ([see below](https://bliptext.com/articles/)) of Lis(e) about μ = 0.\nThe derivatives of the polylogarithm follow from the defining power series:\nThe square relationship is seen from the series definition, and is related to the [duplication formula](https://bliptext.com/articles/duplication-formula) (see also, ):\n[Kummer's function](https://bliptext.com/articles/kummer-s-function) obeys a very similar duplication formula.\nThis is a special case of the [multiplication formula](https://bliptext.com/articles/multiplication-formula), for any positive integer p:\nwhich can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform)).\nAnother important property, the inversion formula, involves the [Hurwitz zeta function](https://bliptext.com/articles/hurwitz-zeta-function) or the [Bernoulli polynomials](https://bliptext.com/articles/bernoulli-polynomials) and is found under [relationship to other functions](https://bliptext.com/articles/) below.\n\n# Particular values\nFor particular cases, the polylogarithm may be expressed in terms of other functions ([see below](https://bliptext.com/articles/)).\nParticular values for the polylogarithm may thus also be found as particular values of these other functions.\n

\nFor integer values of the polylogarithm order, the following explicit expressions are obtained by repeated application of z·∂/∂z to Li1(z):\nAccordingly the polylogarithm reduces to a ratio of polynomials in z, and is therefore a [rational function](https://bliptext.com/articles/rational-function) of z, for all nonpositive integer orders.\nThe general case may be expressed as a finite sum:\nwhere S(n,k) are the [Stirling numbers of the second kind](https://bliptext.com/articles/stirling-numbers-of-the-second-kind).\nEquivalent formulae applicable to negative integer orders are :\nand:\nwhere are the [Eulerian numbers](https://bliptext.com/articles/eulerian-numbers).\nAll roots of Li−n(z) are distinct and real; they include z = 0, while the remainder is negative and centered about z = −1 on a logarithmic scale.\nAs n becomes large, the numerical evaluation of these rational expressions increasingly suffers from cancellation ; full accuracy can be obtained, however, by computing Li−n(z) via the general relation with the Hurwitz zeta function ([see below](https://bliptext.com/articles/)).\n
\nSome particular expressions for half-integer values of the argument z are:\nwhere ζ is the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function).\nNo formulae of this type are known for higher integer orders, but one has for instance :\nwhich involves the alternating double sum\nIn general one has for integer orders n ≥ 2 :\nwhere ζ(s1, …, sk) is the [multiple zeta function](https://bliptext.com/articles/multiple-zeta-function); for example:\n
\nAs a straightforward consequence of the series definition, values of the polylogarithm at the pth complex [roots of unity](https://bliptext.com/articles/roots-of-unity) are given by the [Fourier sum](https://bliptext.com/articles/discrete-fourier-transform):\nwhere ζ is the [Hurwitz zeta function](https://bliptext.com/articles/hurwitz-zeta-function).\nFor Re(s) > 1, where Lis(1) is finite, the relation also holds with m = 0 or m = p.\nWhile this formula is not as simple as that implied by the more general relation with the Hurwitz zeta function listed under [relationship to other functions](https://bliptext.com/articles/) below, it has the advantage of applying to non-negative integer values of s as well.\nAs usual, the relation may be inverted to express ζ(s, m⁄p) for any m = 1, …, p as a Fourier sum of Lis(exp(2πi k⁄p)) over k = 1, …, p.\n

\n# [polylogarithm | upload.wikimedia.org/wikipedia/commons/6/66/Polylogarithm///plot///negative.svg]\n\n# Relationship to other functions\nz = 1\n, the polylogarithm reduces to the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function)\nη(s)\nis the Dirichlet eta function.\nFor pure imaginary arguments, we have: where\nβ(s)\nis the Dirichlet beta function.\nΓ(1 − s)\n, and at\ns = 0\nby a pole of both zeta functions; a derivation of this formula is given under [series representations](https://bliptext.com/articles/) below.\nWith a little help from a functional equation for the Hurwitz zeta function, the polylogarithm is consequently also related to that function via : which relation holds for\n0 ≤ Re(x) < 1\nif\nIm(x) ≥ 0\n, and for\n0 < Re(x) ≤ 1\nif\nIm(x) < 0\n. Equivalently, for all complex s and for complex\nz ∉\n, the inversion formula reads and for all complex s and for complex\nz ∉\nFor\nz ∉\n, one has\nln(−z) = −ln(−1⁄z)\n, and both expressions agree.\nThese relations furnish the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series.\n(The corresponding equation of and is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously.) See the next item for a simplified formula when s is an integer.\nζ(1−n, x) = −Bn(x) / n\n, and Jonquière's inversion formula for n = 1, 2, 3, … becomes: where again 0 ≤ Re(x) < 1 if Im(x) ≥ 0, and 0 < Re(x) ≤ 1 if Im(x) < 0.\nUpon restriction of the polylogarithm argument to the unit circle, Im(x) = 0, the left hand side of this formula simplifies to 2 Re(Lin(e2πix)) if n is even, and to 2i Im(Lin(e2πix)) if n is odd.\nFor negative integer orders, on the other hand, the divergence of Γ(s) implies for all z that : More generally, one has for\nn = 0, ±1, ±2, ±3, …\nz ∉\n. (The corresponding equation of and is again not correct.)\nTis(z)\ncan be expressed in terms of polylogarithms: The relation in particular implies: which explains the function name.\n\n# Integral representations\nAny of the following integral representations furnishes the [analytic continuation](https://bliptext.com/articles/analytic-continuation) of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series.\n

\nThe polylogarithm can be expressed in terms of the integral of the [Bose–Einstein distribution](https://bliptext.com/articles/bose-einstein-distribution):\nThis converges for Re(s) > 0 and all z except for z real and ≥ 1.\nThe polylogarithm in this context is sometimes referred to as a Bose integral but more commonly as a **Bose–Einstein integral**.\nSimilarly, the polylogarithm can be expressed in terms of the integral of the [Fermi–Dirac distribution](https://bliptext.com/articles/fermi-dirac-distribution):\nThis converges for\nRe(s) > 0\nand all z except for z real and ≤ −1.\nThe polylogarithm in this context is sometimes referred to as a Fermi integral or a [Fermi–Dirac integral](https://bliptext.com/articles/complete-fermi-dirac-integral).\nThese representations are readily verified by [Taylor expansion](https://bliptext.com/articles/taylor-series) of the integrand with respect to z and termwise integration.\nThe papers of Dingle contain detailed investigations of both types of integrals.\nThe polylogarithm is also related to the integral of the [Maxwell–Boltzmann distribution](https://bliptext.com/articles/maxwell-boltzmann-distribution):\nThis also gives the [asymptotic behavior](https://bliptext.com/articles/asymptotic-behavior) of polylogarithm at the vicinity of origin.\n
\nA complementary integral representation applies to Re(s) < 0 and to all z except to z real and ≥ 0:\nThis integral follows from the general relation of the polylogarithm with the [Hurwitz zeta function](https://bliptext.com/articles/hurwitz-zeta-function) ([see above](https://bliptext.com/articles/)) and a familiar integral representation of the latter.\n
\nThe polylogarithm may be quite generally represented by a [Hankel contour](https://bliptext.com/articles/hankel-contour) integral, which extends the Bose–Einstein representation to negative orders s.\nAs long as the t = μ [pole](https://bliptext.com/articles/pole-complex-analysis) of the integrand does not lie on the non-negative real axis, and s ≠ 1, 2, 3, …, we have:\nwhere H represents the Hankel contour.\nThe integrand has a cut along the real axis from zero to infinity, with the axis belonging to the lower half plane of t.\nThe integration starts at +∞ on the upper half plane (Im(t) > 0), circles the origin without enclosing any of the poles t = μ + 2kπi, and terminates at +∞ on the lower half plane (Im(t) < 0).\nFor the case where μ is real and non-negative, we can simply subtract the contribution of the enclosed t = μ pole:\nwhere R is the [residue](https://bliptext.com/articles/residue-complex-analysis) of the pole:\n
\nWhen the [Abel–Plana formula](https://bliptext.com/articles/abel-plana-formula) is applied to the defining series of the polylogarithm, a [Hermite](https://bliptext.com/articles/charles-hermite)-type integral representation results that is valid for all complex z and for all complex s:\nwhere Γ is the [upper incomplete gamma-function](https://bliptext.com/articles/incomplete-gamma-function).\nAll (but not part) of the ln(z) in this expression can be replaced by −ln(1⁄z).\nA related representation which also holds for all complex s,\navoids the use of the incomplete gamma function, but this integral fails for z on the positive real axis if Re(s) ≤ 0.\nThis expression is found by writing 2s Lis(−z) / (−z) = Φ(z2, s, 1⁄2) − z Φ(z2, s, 1), where Φ is the [Lerch transcendent](https://bliptext.com/articles/lerch-transcendent), and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e2πt + 1) in place of 1 / (e2πt − 1) to the second Φ series.\n
\nWe can express an integral for the polylogarithm by integrating the ordinary [geometric series](https://bliptext.com/articles/geometric-series) termwise for as\n

\n\n# Series representations\n

\nAs noted under [integral representations](https://bliptext.com/articles/) above, the Bose–Einstein integral representation of the polylogarithm may be extended to negative orders s by means of [Hankel contour](https://bliptext.com/articles/hankel-contour) integration:\nwhere H is the Hankel contour, s ≠ 1, 2, 3, …, and the t = μ pole of the integrand does not lie on the non-negative real axis.\nThe [contour](https://bliptext.com/articles/methods-of-contour-integration) can be modified so that it encloses the [poles](https://bliptext.com/articles/pole-complex-analysis) of the integrand at t − μ = 2kπi, and the integral can be evaluated as the sum of the [residues](https://bliptext.com/articles/residue-complex-analysis) :\nThis will hold for Re(s) < 0 and all μ except where eμ = 1.\nFor 0 < Im(μ) ≤ 2π the sum can be split as:\nwhere the two series can now be identified with the [Hurwitz zeta function](https://bliptext.com/articles/hurwitz-zeta-function):\nThis relation, which has already been given under [relationship to other functions](https://bliptext.com/articles/) above, holds for all complex s ≠ 0, 1, 2, 3, … and was first derived in.\n
\nIn order to represent the polylogarithm as a power series about μ = 0, we write the series derived from the Hankel contour integral as:\nWhen the binomial powers in the sum are expanded about μ = 0 and the order of summation is reversed, the sum over h can be expressed in closed form:\nThis result holds for |μ| < 2π and, thanks to the analytic continuation provided by the [zeta functions](https://bliptext.com/articles/riemann-zeta-function), for all s ≠ 1, 2, 3, ….\nIf the order is a positive integer, s = n, both the term with k = n − 1 and the [gamma function](https://bliptext.com/articles/gamma-function) become infinite, although their sum does not.\nOne obtains :\nwhere the sum over h vanishes if k = 0.\nSo, for positive integer orders and for |μ| < 2π we have the series:\nwhere Hn denotes the nth [harmonic number](https://bliptext.com/articles/harmonic-number):\nThe problem terms now contain −ln(−μ) which, when multiplied by μn−1, will tend to zero as μ → 0, except for n = 1.\nThis reflects the fact that Lis(z) exhibits a true [logarithmic singularity](https://bliptext.com/articles/mathematical-singularity) at s = 1 and z = 1 since:\nFor s close, but not equal, to a positive integer, the divergent terms in the expansion about μ = 0 can be expected to cause computational difficulties.\nErdélyi's corresponding expansion in powers of ln(z) is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously, since ln(1⁄z) is not uniformly equal to −ln(z).\nFor nonpositive integer values of s, the zeta function ζ(s − k) in the expansion about μ = 0 reduces to [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers): ζ(−n − k) = −B1+n+k / (1 + n + k).\nNumerical evaluation of Li−n(z) by this series does not suffer from the cancellation effects that the finite rational expressions given under [particular values](https://bliptext.com/articles/) above exhibit for large n.\n
\nBy use of the identity\nthe Bose–Einstein integral representation of the polylogarithm ([see above](https://bliptext.com/articles/)) may be cast in the form:\nReplacing the hyperbolic cotangent with a bilateral series,\nthen reversing the order of integral and sum, and finally identifying the summands with an integral representation of the [upper incomplete gamma function](https://bliptext.com/articles/incomplete-gamma-function), one obtains:\nFor both the bilateral series of this result and that for the hyperbolic cotangent, symmetric partial sums from −kmax to kmax converge unconditionally as kmax → ∞.\nProvided the summation is performed symmetrically, this series for Lis(z) thus holds for all complex s as well as all complex z.\n
\nIntroducing an explicit expression for the [Stirling numbers of the second kind](https://bliptext.com/articles/stirling-numbers-of-the-second-kind) into the finite sum for the polylogarithm of nonpositive integer order ([see above](https://bliptext.com/articles/)) one may write:\nThe infinite series obtained by simply extending the outer summation to ∞ :\nturns out to converge to the polylogarithm for all complex s and for complex z with Re(z) < 1⁄2, as can be verified for |−z⁄(1−z)| < 1⁄2 by reversing the order of summation and using:\nThe inner coefficients of these series can be expressed by [Stirling-number-related](https://bliptext.com/articles/stirling-numbers-of-the-first-kind) formulas involving the generalized [harmonic numbers](https://bliptext.com/articles/harmonic-numbers).\nFor example, see [generating function transformations](https://bliptext.com/articles/generating-function-transformation) to find proofs (references to proofs) of the following identities:\nFor the other arguments with Re(z) < 1⁄2 the result follows by [analytic continuation](https://bliptext.com/articles/analytic-continuation).\nThis procedure is equivalent to applying [Euler's transformation](https://bliptext.com/articles/binomial-transform) to the series in z that defines the polylogarithm.\n

\n\n# Asymptotic expansions\nFor |z| ≫ 1, the polylogarithm can be expanded into [asymptotic series](https://bliptext.com/articles/asymptotic-expansion) in terms of ln(−z):\nwhere B2k are the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers).\nBoth versions hold for all s and for any arg(z).\nAs usual, the summation should be terminated when the terms start growing in magnitude.\nFor negative integer s, the expansions vanish entirely; for non-negative integer s, they break off after a finite number of terms.\ndescribes a method for obtaining these series from the Bose–Einstein integral representation (his equation 11.2 for Lis(eμ) requires −2π < Im(μ) ≤ 0).\n\n# Limiting behavior\nThe following [limits](https://bliptext.com/articles/limit-mathematics) result from the various representations of the polylogarithm :\nWood's first limit for\nRe(μ) → ∞\nhas been corrected in accordance with his equation 11.3.\nThe limit for\nRe(s) → −∞\nfollows from the general relation of the polylogarithm with the [Hurwitz zeta function](https://bliptext.com/articles/hurwitz-zeta-function) ([see above](https://bliptext.com/articles/)).\n\n# Dilogarithm\nThe dilogarithm is the polylogarithm of order s = 2.\nAn alternate integral expression of the dilogarithm for arbitrary complex argument z is :\nA source of confusion is that some [computer algebra systems](https://bliptext.com/articles/computer-algebra-system) define the dilogarithm as dilog(z) = Li2(1−z).\nIn the case of real z ≥ 1 the first integral expression for the dilogarithm can be written as\nfrom which expanding ln(t−1) and integrating term by term we obtain\nThe [Abel](https://bliptext.com/articles/niels-henrik-abel) identity for the dilogarithm is given by\nThis is immediately seen to hold for either x = 0 or y = 0, and for general arguments is then easily verified by differentiation ∂/∂x ∂/∂y.\nFor y = 1−x the identity reduces to [Euler](https://bliptext.com/articles/leonhard-euler)'s reflection formula\nwhere Li2(1) = ζ(2) = 1⁄6 π2 has been used and x may take any complex value.\nIn terms of the new variables u = x/(1−y), v = y/(1−x) the Abel identity reads\nwhich corresponds to the pentagon identity given in.\nFrom the Abel identity for x = y = 1−z and the square relationship we have [Landen](https://bliptext.com/articles/john-landen)'s identity\nand applying the reflection formula to each dilogarithm we find the inversion formula\nand for real z ≥ 1 also\nKnown closed-form evaluations of the dilogarithm at special arguments are collected in the table below.\nArguments in the first column are related by reflection x ↔ 1−x or inversion x ↔ 1⁄x to either x = 0 or x = −1; arguments in the third column are all interrelated by these operations.\ndiscusses the 17th to 19th century references.\nThe reflection formula was already published by Landen in 1760, prior to its appearance in a 1768 book by Euler ; an equivalent to Abel's identity was already published by [Spence](https://bliptext.com/articles/william-spence-mathematician) in 1809, before Abel wrote his manuscript in 1826.\nThe designation bilogarithmische Function was introduced by [Carl Johan Danielsson Hill](https://bliptext.com/articles/sv-carl-johan-hill) (professor in Lund, Sweden) in 1828.\nhas remarked that the dilogarithm is the only mathematical function possessing a sense of humor.\n! x !\n! x !\n\n# Polylogarithm ladders\n[Leonard Lewin](https://bliptext.com/articles/leonard-lewin-telecommunications-engineer) discovered a remarkable and broad generalization of a number of classical relationships on the polylogarithm for special values.\nThese are now called polylogarithm ladders.\nDefine as the reciprocal of the [golden ratio](https://bliptext.com/articles/golden-ratio).\nThen two simple examples of dilogarithm ladders are\ngiven by and\ngiven by [Landen](https://bliptext.com/articles/john-landen).\nPolylogarithm ladders occur naturally and deeply in [K-theory](https://bliptext.com/articles/k-theory) and [algebraic geometry](https://bliptext.com/articles/algebraic-geometry).\nPolylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the [BBP algorithm](https://bliptext.com/articles/bbp-algorithm).\n\n# Monodromy\nThe polylogarithm has two [branch points](https://bliptext.com/articles/branch-point); one at z = 1 and another at z = 0.\nThe second branch point, at z = 0, is not visible on the main sheet of the polylogarithm; it becomes visible only when the function is [analytically continued](https://bliptext.com/articles/analytically-continued) to its other sheets.\nThe [monodromy](https://bliptext.com/articles/monodromy) group for the polylogarithm consists of the [homotopy](https://bliptext.com/articles/homotopy) classes of loops that wind around the two branch points.\nDenoting these two by m0 and m1, the monodromy group has the [group presentation](https://bliptext.com/articles/group-presentation)\nFor the special case of the dilogarithm, one also has that wm0 = m0w, and the monodromy group becomes the [Heisenberg group](https://bliptext.com/articles/heisenberg-group) (identifying m0, m1 and w with x, y, z).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:45:57.617Z","updated_at":"2025-01-25T19:45:57.617Z","search_vector":"'polylogarithm':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **polylogarithm** (also known as **Jonquière's function**, for Alfred Jonquière) is a [special function](https://bliptext.com/articles/special-function)\nLis(z)\nof order s and argument z. Only for special values of s does the polylogarithm reduce to an [elementary function](https://bliptext.com/articles/elementary-function) such as the [natural logarithm](https://bliptext.com/articles/natural-logarithm) or a [rational function](https://bliptext.com/articles/rational-function).\nIn [quantum statistics](https://bliptext.com/articles/quantum-statistics), the polylogarithm function appears as the closed form of [integrals](https://bliptext.com/articles/integral) of the [Fermi–Dirac distribution](https://bliptext.com/articles/fermi-dirac-distribution) and the [Bose–Einstein distribution](https://bliptext.com/articles/bose-einstein-distribution), and is also known as the **Fermi–Dirac integral** or the **Bose–Einstein integral**.\nIn [quantum electrodynamics](https://bliptext.com/articles/quantum-electrodynamics), polylogarithms of positive [integer](https://bliptext.com/articles/integer) order arise in the calculation of processes represented by higher-order [Feynman diagrams](https://bliptext.com/articles/feynman-diagram).\nThe polylogarithm [function](https://bliptext.com/articles/function-mathematics) is equivalent to the [Hurwitz zeta [function](https://bliptext.com/articles/function-mathematics)](https://bliptext.com/articles/hurwitz-zeta-[function](https://bliptext.com/articles/function-mathematics)) — either [function](https://bliptext.com/articles/function-mathematics) can be expressed in terms of the other — and both [function](https://bliptext.com/articles/function-mathematics)s are special cases of the [Lerch transcendent](https://bliptext.com/articles/lerch-transcendent).\nPolylogarithms should not be confused with [polylogarithmic functions](https://bliptext.com/articles/polylogarithmic-function), nor with the [offset logarithmic integral](https://bliptext.com/articles/offset-logarithmic-integral)\nLi(z)\n, which has the same notation without the subscript.\nThe polylogarithm function is defined by a [power series](https://bliptext.com/articles/power-series) in z, which is also a [Dirichlet series](https://bliptext.com/articles/dirichlet-series) in s:\nThis definition is valid for arbitrary [complex](https://bliptext.com/articles/complex-numbers) order s and for all [complex](https://bliptext.com/articles/complex-numbers) arguments z with\nby the process of [analytic continuation](https://bliptext.com/articles/analytic-continuation).\n(Here the denominator ks is understood as\nexp(s ln k)\n). The special case\ns = 1\ninvolves the ordinary [natural logarithm](https://bliptext.com/articles/natural-logarithm),\nLi1(z) = −ln(1−z)\n, while the special cases\ns = 2\nand\ns = 3\nare called the [dilogarithm](https://bliptext.com/articles/dilogarithm) (also referred to as Spence's function) and trilogarithm respectively.\nThe name of the function comes from the fact that it may also be defined as the repeated [integral](https://bliptext.com/articles/indefinite-integral) of itself:\nthus the dilogarithm is an integral of a function involving the logarithm, and so on.\nFor nonpositive integer orders s, the polylogarithm is a [rational function](https://bliptext.com/articles/rational-function).\n\n# Properties\nIn the case where the order s is an integer, it will be represented by s=n (or s=-n when negative).\nIt is often convenient to define \\mu=\\ln(z) where \\ln(z) is the [principal branch](https://bliptext.com/articles/principal-branch) of the [complex logarithm](https://bliptext.com/articles/complex-logarithm) so that Also, all exponentiation will be assumed to be single-valued:\nDepending on the order s, the polylogarithm may be multi-valued.\nThe principal branch of is taken to be given for | z | \n \nFor integer values of the polylogarithm order, the following explicit expressions are obtained by repeated application of z·∂/∂z to Li1(z):\nAccordingly the polylogarithm reduces to a ratio of polynomials in z, and is therefore a [rational function](https://bliptext.com/articles/rational-function) of z, for all nonpositive integer orders.\nThe general case may be expressed as a finite sum:\nwhere S(n,k) are the [Stirling numbers of the second kind](https://bliptext.com/articles/stirling-numbers-of-the-second-kind).\nEquivalent formulae applicable to negative integer orders are :\nand:\nwhere are the [Eulerian numbers](https://bliptext.com/articles/eulerian-numbers).\nAll roots of Li−n(z) are distinct and real; they include z = 0, while the remainder is negative and centered about z = −1 on a logarithmic scale.\nAs n becomes large, the numerical evaluation of these rational expressions increasingly suffers from cancellation ; full accuracy can be obtained, however, by computing Li−n(z) via the general relation with the Hurwitz zeta function ([see below](https://bliptext.com/articles/)).\n \n \nSome particular expressions for half-integer values of the argument z are:\nwhere ζ is the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function).\nNo formulae of this type are known for higher integer orders, but one has for instance :\nwhich involves the alternating double sum\nIn general one has for integer orders n ≥ 2 :\nwhere ζ(s1, …, sk) is the [multiple zeta function](https://bliptext.com/articles/multiple-zeta-function); for example:\n \n \nAs a straightforward consequence of the series definition, values of the polylogarithm at the pth complex [roots of unity](https://bliptext.com/articles/roots-of-unity) are given by the [Fourier sum](https://bliptext.com/articles/discrete-fourier-transform):\nwhere ζ is the [Hurwitz zeta function](https://bliptext.com/articles/hurwitz-zeta-function).\nFor Re(s) > 1, where Lis(1) is finite, the relation also holds with m = 0 or m = p.\nWhile this formula is not as simple as that implied by the more general relation with the Hurwitz zeta function listed under [relationship to other functions](https://bliptext.com/articles/) below, it has the advantage of applying to non-negative integer values of s as well.\nAs usual, the relation may be inverted to express ζ(s, m⁄p) for any m = 1, …, p as a Fourier sum of Lis(exp(2πi k⁄p)) over k = 1, …, p.\n \n \n# [polylogarithm | upload.wikimedia.org/wikipedia/commons/6/66/Polylogarithm///plot///negative.svg]\n\n# Relationship to other functions\nz = 1\n, the polylogarithm reduces to the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function)\nη(s)\nis the Dirichlet eta function.\nFor pure imaginary arguments, we have: where\nβ(s)\nis the Dirichlet beta function.\nΓ(1 − s)\n, and at\ns = 0\nby a pole of both zeta functions; a derivation of this formula is given under [series representations](https://bliptext.com/articles/) below.\nWith a little help from a functional equation for the Hurwitz zeta function, the polylogarithm is consequently also related to that function via : which relation holds for\n0 ≤ Re(x) \n \nThe polylogarithm can be expressed in terms of the integral of the [Bose–Einstein distribution](https://bliptext.com/articles/bose-einstein-distribution):\nThis converges for Re(s) > 0 and all z except for z real and ≥ 1.\nThe polylogarithm in this context is sometimes referred to as a Bose integral but more commonly as a **Bose–Einstein integral**.\nSimilarly, the polylogarithm can be expressed in terms of the integral of the [Fermi–Dirac distribution](https://bliptext.com/articles/fermi-dirac-distribution):\nThis converges for\nRe(s) > 0\nand all z except for z real and ≤ −1.\nThe polylogarithm in this context is sometimes referred to as a Fermi integral or a [Fermi–Dirac integral](https://bliptext.com/articles/complete-fermi-dirac-integral).\nThese representations are readily verified by [Taylor expansion](https://bliptext.com/articles/taylor-series) of the integrand with respect to z and termwise integration.\nThe papers of Dingle contain detailed investigations of both types of integrals.\nThe polylogarithm is also related to the integral of the [Maxwell–Boltzmann distribution](https://bliptext.com/articles/maxwell-boltzmann-distribution):\nThis also gives the [asymptotic behavior](https://bliptext.com/articles/asymptotic-behavior) of polylogarithm at the vicinity of origin.\n \n \nA complementary integral representation applies to Re(s) \n \nThe polylogarithm may be quite generally represented by a [Hankel contour](https://bliptext.com/articles/hankel-contour) integral, which extends the Bose–Einstein representation to negative orders s.\nAs long as the t = μ [pole](https://bliptext.com/articles/pole-complex-analysis) of the integrand does not lie on the non-negative real axis, and s ≠ 1, 2, 3, …, we have:\nwhere H represents the Hankel contour.\nThe integrand has a cut along the real axis from zero to infinity, with the axis belonging to the lower half plane of t.\nThe integration starts at +∞ on the upper half plane (Im(t) > 0), circles the origin without enclosing any of the poles t = μ + 2kπi, and terminates at +∞ on the lower half plane (Im(t) \n \nWhen the [Abel–Plana formula](https://bliptext.com/articles/abel-plana-formula) is applied to the defining series of the polylogarithm, a [Hermite](https://bliptext.com/articles/charles-hermite)-type integral representation results that is valid for all complex z and for all complex s:\nwhere Γ is the [upper incomplete gamma-function](https://bliptext.com/articles/incomplete-gamma-function).\nAll (but not part) of the ln(z) in this expression can be replaced by −ln(1⁄z).\nA related representation which also holds for all complex s,\navoids the use of the incomplete gamma function, but this integral fails for z on the positive real axis if Re(s) ≤ 0.\nThis expression is found by writing 2s Lis(−z) / (−z) = Φ(z2, s, 1⁄2) − z Φ(z2, s, 1), where Φ is the [Lerch transcendent](https://bliptext.com/articles/lerch-transcendent), and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e2πt + 1) in place of 1 / (e2πt − 1) to the second Φ series.\n \n \nWe can express an integral for the polylogarithm by integrating the ordinary [geometric series](https://bliptext.com/articles/geometric-series) termwise for as\n \n \n\n# Series representations\n \n \nAs noted under [integral representations](https://bliptext.com/articles/) above, the Bose–Einstein integral representation of the polylogarithm may be extended to negative orders s by means of [Hankel contour](https://bliptext.com/articles/hankel-contour) integration:\nwhere H is the Hankel contour, s ≠ 1, 2, 3, …, and the t = μ pole of the integrand does not lie on the non-negative real axis.\nThe [contour](https://bliptext.com/articles/methods-of-contour-integration) can be modified so that it encloses the [poles](https://bliptext.com/articles/pole-complex-analysis) of the integrand at t − μ = 2kπi, and the integral can be evaluated as the sum of the [residues](https://bliptext.com/articles/residue-complex-analysis) :\nThis will hold for Re(s) \n \nIn order to represent the polylogarithm as a power series about μ = 0, we write the series derived from the Hankel contour integral as:\nWhen the binomial powers in the sum are expanded about μ = 0 and the order of summation is reversed, the sum over h can be expressed in closed form:\nThis result holds for |μ| \n \nBy use of the identity\nthe Bose–Einstein integral representation of the polylogarithm ([see above](https://bliptext.com/articles/)) may be cast in the form:\nReplacing the hyperbolic cotangent with a bilateral series,\nthen reversing the order of integral and sum, and finally identifying the summands with an integral representation of the [upper incomplete gamma function](https://bliptext.com/articles/incomplete-gamma-function), one obtains:\nFor both the bilateral series of this result and that for the hyperbolic cotangent, symmetric partial sums from −kmax to kmax converge unconditionally as kmax → ∞.\nProvided the summation is performed symmetrically, this series for Lis(z) thus holds for all complex s as well as all complex z.\n \n \nIntroducing an explicit expression for the [Stirling numbers of the second kind](https://bliptext.com/articles/stirling-numbers-of-the-second-kind) into the finite sum for the polylogarithm of nonpositive integer order ([see above](https://bliptext.com/articles/)) one may write:\nThe infinite series obtained by simply extending the outer summation to ∞ :\nturns out to converge to the polylogarithm for all complex s and for complex z with Re(z) \n \n\n# Asymptotic expansions\nFor |z| ≫ 1, the polylogarithm can be expanded into [asymptotic series](https://bliptext.com/articles/asymptotic-expansion) in terms of ln(−z):\nwhere B2k are the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers).\nBoth versions hold for all s and for any arg(z).\nAs usual, the summation should be terminated when the terms start growing in magnitude.\nFor negative integer s, the expansions vanish entirely; for non-negative integer s, they break off after a finite number of terms.\ndescribes a method for obtaining these series from the Bose–Einstein integral representation (his equation 11.2 for Lis(eμ) requires −2π < Im(μ) ≤ 0).\n\n# Limiting behavior\nThe following [limits](https://bliptext.com/articles/limit-mathematics) result from the various representations of the polylogarithm :\nWood's first limit for\nRe(μ) → ∞\nhas been corrected in accordance with his equation 11.3.\nThe limit for\nRe(s) → −∞\nfollows from the general relation of the polylogarithm with the [Hurwitz zeta function](https://bliptext.com/articles/hurwitz-zeta-function) ([see above](https://bliptext.com/articles/)).\n\n# Dilogarithm\nThe dilogarithm is the polylogarithm of order s = 2.\nAn alternate integral expression of the dilogarithm for arbitrary complex argument z is :\nA source of confusion is that some [computer algebra systems](https://bliptext.com/articles/computer-algebra-system) define the dilogarithm as dilog(z) = Li2(1−z).\nIn the case of real z ≥ 1 the first integral expression for the dilogarithm can be written as\nfrom which expanding ln(t−1) and integrating term by term we obtain\nThe [Abel](https://bliptext.com/articles/niels-henrik-abel) identity for the dilogarithm is given by\nThis is immediately seen to hold for either x = 0 or y = 0, and for general arguments is then easily verified by differentiation ∂/∂x ∂/∂y.\nFor y = 1−x the identity reduces to [Euler](https://bliptext.com/articles/leonhard-euler)'s reflection formula\nwhere Li2(1) = ζ(2) = 1⁄6 π2 has been used and x may take any complex value.\nIn terms of the new variables u = x/(1−y), v = y/(1−x) the Abel identity reads\nwhich corresponds to the pentagon identity given in.\nFrom the Abel identity for x = y = 1−z and the square relationship we have [Landen](https://bliptext.com/articles/john-landen)'s identity\nand applying the reflection formula to each dilogarithm we find the inversion formula\nand for real z ≥ 1 also\nKnown closed-form evaluations of the dilogarithm at special arguments are collected in the table below.\nArguments in the first column are related by reflection x ↔ 1−x or inversion x ↔ 1⁄x to either x = 0 or x = −1; arguments in the third column are all interrelated by these operations.\ndiscusses the 17th to 19th century references.\nThe reflection formula was already published by Landen in 1760, prior to its appearance in a 1768 book by Euler ; an equivalent to Abel's identity was already published by [Spence](https://bliptext.com/articles/william-spence-mathematician) in 1809, before Abel wrote his manuscript in 1826.\nThe designation bilogarithmische Function was introduced by [Carl Johan Danielsson Hill](https://bliptext.com/articles/sv-carl-johan-hill) (professor in Lund, Sweden) in 1828.\nhas remarked that the dilogarithm is the only mathematical function possessing a sense of humor.\n! x !\n! x !\n\n# Polylogarithm ladders\n[Leonard Lewin](https://bliptext.com/articles/leonard-lewin-telecommunications-engineer) discovered a remarkable and broad generalization of a number of classical relationships on the polylogarithm for special values.\nThese are now called polylogarithm ladders.\nDefine as the reciprocal of the [golden ratio](https://bliptext.com/articles/golden-ratio).\nThen two simple examples of dilogarithm ladders are\ngiven by and\ngiven by [Landen](https://bliptext.com/articles/john-landen).\nPolylogarithm ladders occur naturally and deeply in [K-theory](https://bliptext.com/articles/k-theory) and [algebraic geometry](https://bliptext.com/articles/algebraic-geometry).\nPolylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the [BBP algorithm](https://bliptext.com/articles/bbp-algorithm).\n\n# Monodromy\nThe polylogarithm has two [branch points](https://bliptext.com/articles/branch-point); one at z = 1 and another at z = 0.\nThe second branch point, at z = 0, is not visible on the main sheet of the polylogarithm; it becomes visible only when the function is [analytically continued](https://bliptext.com/articles/analytically-continued) to its other sheets.\nThe [monodromy](https://bliptext.com/articles/monodromy) group for the polylogarithm consists of the [homotopy](https://bliptext.com/articles/homotopy) classes of loops that wind around the two branch points.\nDenoting these two by m0 and m1, the monodromy group has the [group presentation](https://bliptext.com/articles/group-presentation)\nFor the special case of the dilogarithm, one also has that wm0 = m0w, and the monodromy group becomes the [Heisenberg group](https://bliptext.com/articles/heisenberg-group) (identifying m0, m1 and w with x, y, z).","revisionCount":0},"hasMore":false,"nextCursor":null}

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