The dilogarithm

The dilogarithm

is analytic on \mathbb{C} \setminus [1, \infty). acb_hypgeom_dilog evaluates it for complex balls; it is a holonomic function, and the methods below are instances of the general ones in Hypergeometric functions. General polylogarithms \operatorname{Li}_s(z) are computed in acb_dirichlet and acb_poly by other methods (Zeta functions, polylogarithms and L-functions).

Transformations#

The functional equations

map z to 1/z, 1 - z, z/(z-1) and 1/(1-z) (acb_hypgeom_dilog_transform with algorithm 1 to 4; each returns an indeterminate result if the ball meets the region where the formula is invalid). As for {}_2F_1, the union of the disks |w| \le r over the transformed arguments w covers the plane except neighbourhoods of the points e^{\pm\pi i/3}, which are fixed points of the transformation group.

acb_hypgeom_dilog computes |z|^2, |1/z|^2, |1-z|^2, |z/(z-1)|^2 and |1/(1-z)|^2 in double precision and uses the series at 0 directly if |z| \le 1/4, and otherwise after the transformation that minimizes |w|. If the best |w| is still above 1/4, it considers two alternatives:

• below 10000 bits, the series in \log z described next, if its rate of convergence |\log z/(2\pi)| (with a penalty factor) is better;
• if the best |w| exceeds 3/4 (which happens only near e^{\pm\pi i/3} when the Bernoulli series was not chosen), expansion at one of the points i, (1+i)/2, 1+i (or their conjugates), where \operatorname{Li}_2 has a closed form, for example \operatorname{Li}_2(i) = -\pi^2/48 + iG with Catalan's constant G, followed by bit-burst continuation to z.

The working precision is limited to the accuracy of the input plus 30 bits, and raised by 0.5% plus 5 bits.

The series at zero#

acb_hypgeom_dilog_zero_taylor sums \sum_{k \ge 1} z^k/k^2 for small |z|. Since the coefficients are reciprocals of small integers, the sum is computed by rectangular splitting in powers of z, where several consecutive denominators k^2 are combined into one word-size divisor, or by binary splitting when z has few bits; the tail after n terms is bounded by |z|^n/(1-|z|). acb_hypgeom_dilog_zero uses this below 40000 bits or for tiny |z|, and the bit-burst algorithm otherwise.

The series in log z#

With w = \log z, for |w| < 2\pi,

which follows from \operatorname{Li}_2(e^w) = \zeta(2) + w(1 - \log(-w)) + \sum_{k \ge 2} \zeta(2-k) w^k/k! and \zeta(2-k) = -B_{k-1}/(k-1). For k \ge 4 the terms are bounded by (|w|/(2\pi))^k, which gives the truncation bound as a geometric series. acb_hypgeom_dilog_bernoulli evaluates this with Horner's scheme in w^2 and the cached Bernoulli numbers. Near z = e^{\pm\pi i/3}, |w| \approx \pi/3 and each term gains about 2.6 bits. Since the Bernoulli numbers grow in size, the method is used only below 10000 bits.

Taylor continuation#

acb_hypgeom_dilog_continuation computes \operatorname{Li}_2(z) - \operatorname{Li}_2(a) from the Taylor expansion at a. Since \frac{d}{dz}\operatorname{Li}_2(z) = -\log(1-z)/z, the Taylor coefficients c_k satisfy a recurrence of order 2 with polynomial coefficients in k, a and a - 1, so the partial sums are products of 3 \times 3 matrices, computed with the forward recurrence or by binary splitting. The coefficients are bounded by

which follows from the symbolic derivatives, or by |c_k| \le 4/|1-a|^k for |a| \le 1/2 and a = \pm i (from Cauchy's integral formula); the tail after n terms is then bounded by a geometric series in |z - a|/\min(|a|, |1-a|), which must be less than 1. If the segment from a to z crosses the branch cut, the result is the analytic continuation along the segment.

The bit-burst algorithm#

For a full-precision z, the Taylor series at a needs O(p) terms with full-precision coefficients. acb_hypgeom_dilog_bitburst (The bit-burst algorithm) instead chooses z_0 as z truncated to 16 fractional bits and the points z_j as z truncated to 2^{j+4} bits, and sums

each term by Taylor continuation from z_j with binary splitting: the argument z_{j+1} - z_j is small and short, so each step costs O(M(p)\log p) and the total O(M(p)\log^2 p). The value \operatorname{Li}_2(z_0) comes from the series at 0 (or from a continuation from a special point), and since z_0 has only 16 bits it is cheap. If the path would come close to the branch cut, the bit-burst step is skipped.

Use in FLINT#

• acb_hypgeom_dilog and the real arb_hypgeom_dilog; the methods acb_hypgeom_dilog_zero, acb_hypgeom_dilog_zero_taylor, acb_hypgeom_dilog_transform, acb_hypgeom_dilog_bernoulli, acb_hypgeom_dilog_continuation, acb_hypgeom_dilog_bitburst.
• acb_polylog with s = 2 (through _acb_poly_polylog_cpx) and the generic gr_dilog for complex ball contexts call acb_hypgeom_dilog.