Polylogarithm and related functions

Polylogarithm, \(\mathrm{Li}_s(z)\)

ctx.polylog(x, s)

where ctx is math53, ctxflint.

Returns the polylogarithm of real order \(s\), \(\displaystyle \mathrm{Li}_s(x) = \sum_{k=1}^{\infty} \frac{x^k}{k^s}, \quad s \ge -1, |x| \le 1 \,\), or its analytic continuation. For \(s\leq 1\) there is the additional arguments restriction \(z\neq 1\).

See also Wikipedia [1438], MathWorld [1043], NIST [19], Ehrhardt [309] (3.6.12), Flint [841], Mpmath [692].

Special cases are \(\mathrm{Li}_s(0) = 0\), \(\mathrm{Li}_s(1) = \zeta(s)\), \(\mathrm{Li}_s(-1) = -\eta(s)\). For \(x>1\) the real part of \(\mathrm{Li}_s(x)\) is returned.

\(\quad\) \(\quad\)

3D wpf plot: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function \(z = \sin(x + iy)\), with \(-6 \le x \le 6\) (blue axis), \(-6 \le y \le 6\) (red axis), \(-10 \le z \le 10\) (green axis). Function values are loglog-transformed.

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.PolylogR(2, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.PolylogR(2, '0.1')
ereal('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; s = '10'; x = '0.5'
>>> \mathrm{d}x = dec.polylog(s, x); mx = mpm.polylog(s, x); gx = gmp.polylog(s, x)
>>> fx = fpm.polylog(s, x); ax = apm.polylog(s, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  5.002463206060067750096752404960275553344E-1
mpm:  5.002463206060067750096752404960275553344e-1
gmp:  5.002463206060067750096752404960275553344E-01
fpm:  5.00246320606007E-01
apm:  5.002463206060067750096752404756715463182e-1 (4.658e-24%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; s = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.polylog(s, z); mz = mpm.polylog(s, z); gz = gmp.polylog(m, z)
>>> fz = fpm.polylog(m, z); az = apm.polylog(m, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 5.0140954368998825423E+0             + 3.0330881889981661901E+0j
mpm: 5.0140954368998825423e+0             + 3.0330881889981661901e+0j
gmp: 5.0140954368998825423E+00            + 3.0330881889981661901E+00j
fpm: 5.01409543689988E+00                 + 3.03308818899817E+00j
apm: 5.0140954366116116588e+0 (1.145e-5%) + 3.0330881892210954918e+0 (2.687e-5%)j

Trilogarithm Function, \(\mathrm{Li}_3(z)\)

math53.trilog(x)

Returns the trilogarithm \(\displaystyle \mathrm{trilog}(x) = \Re \displaystyle \mathrm{Li}_3(x)\). See also Wikipedia [1442], MathWorld [1028], NIST [17], Ehrhardt [309] (3.6.14), Mpmath [692].

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Trilog(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Trilog('0.1')
ereal('5.3518479027559984754E-1')

Dilogarithm Function, \(\mathrm{Li}_2(z)\)

ctx.dilog(x)

where ctx is math53, mathc53 or ctxflint.

Returns the dilogarithm \(\displaystyle \mathrm{dilog}(x) = \Re \displaystyle \mathrm{Li}_2(x) = -\Re \int_0^x \frac{\log(1-t)}{t} \, \mathrm{d}t\).

See also Wikipedia [1442], MathWorld [1028], NIST [17], Ehrhardt [309] (3.6.13), Ehrhardt [309] (4.2.25), Flint [840], Mpmath [692].

Note that there is some confusion about the naming: some authors and/or computer algebra systems use \(\text{dilog}(x) = \text{Li}_2(1-x)\) and then call \(\text{Li}_2(x)\) Spence function/integral or similar.

\(\quad\) \(\quad\)

3D wpf plot: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function \(z = \sin(x + iy)\), with \(-6 \le x \le 6\) (blue axis), \(-6 \le y \le 6\) (red axis), \(-10 \le z \le 10\) (green axis). Function values are loglog-transformed.

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Dilog(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Dilog('0.1')
ereal('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; x = '0.5'
>>> \mathrm{d}x = dec.dilog(x); mx = mpm.dilog(x); gx = gmp.dilog(x)
>>> fx = fpm.dilog(x); ax = apm.dilog(x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  5.822405264650125059026563201596801087442E-1
mpm:  5.822405264650125059026563201596801087442e-1
gmp:  5.822405264650125059026563201596801087442E-01
fpm:  5.82240526465013E-01
apm:  5.822405264650125059026563201596801087442e-1 (9.858e-40%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.dilog(z); mz = mpm.dilog(z); gz = gmp.dilog(z)
>>> fz = fpm.dilog(z); az = apm.dilog(z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 3.3260008208192192025E-2              + 4.6816673723804519905E+0j
mpm: 3.3260008208192192025e-2              + 4.6816673723804519905e+0j
gmp: 3.3260008208192192025E-02             + 4.6816673723804519905E+00j
fpm: 3.32600082081922E-02                  + 4.68166737238045E+00j
apm: 3.3260008208192192025e-2 (7.958e-20%) + 4.6816673723804519905e+0 (7.237e-20%)j

Generalized Clausen sine function

ctxflint.clausen_sin(s, z)

Returns the Clausen sine function. See also Wikipedia [1465], MathWorld [1095], Mpmath [737].

Computes the Clausen sine function, defined formally by the series

\[\mathrm{Cl}_s(z) = \sum_{k=1}^{\infty} \frac{\sin(kz)}{k^s}.\]

The special case \(\mathrm{Cl}_2(z)\) (i.e. clsin(2,z)) is the classical “Clausen function”. More generally, the Clausen function is defined for complex \(s\) and \(z\), even when the series does not converge. The Clausen function is related to the polylogarithm (polylog()) as

\[ \begin{align}\begin{aligned}\mathrm{Cl}_s(z) = \frac{1}{2i}\left(\mathrm{Li}_s\left(e^{iz}\right) - \mathrm{Li}_s\left(e^{-iz}\right)\right)\\= \mathrm{Im}\left[\mathrm{Li}_s(e^{iz})\right] \quad (s, z \in \mathbb{R}),\end{aligned}\end{align} \]

and this representation can be taken to provide the analytic continuation of the series. The complementary function clcos() gives the corresponding cosine sum.

\(\quad\) \(\quad\)

3D wpf plot: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function \(z = \sin(x + iy)\), with \(-6 \le x \le 6\) (blue axis), \(-6 \le y \le 6\) (red axis), \(-10 \le z \le 10\) (green axis). Function values are loglog-transformed.

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; s = '3'; x = '4'
>>> \mathrm{d}x = dec.clsin(s, x); mx = mpm.clsin(s, x); gx = gmp.clsin(s, x)
>>> fx = fpm.clsin(s, x); ax = apm.clsin(s, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  -6.533010136329338746275795332005774465795E-1
mpm:  -6.533010136329338746275795332005774465795e-1
gmp:  -6.533010136329338746275795332005774465795E-01
fpm:  -6.53301013632934E-01
apm:  -6.533010136329338746275795332005774886000e-1 (-2.242e-29%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; s = '3'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.clsin(s, z); mz = mpm.clsin(s, z); gz = gmp.clsin(s, z)
>>> fz = fpm.clsin(s, z); az = apm.clsin(s, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: -5.1750336134513741048E+0               - 2.1271427013787699791E+0j
mpm: -5.1750336134513741048e+0               - 2.1271427013787699791e+0j
gmp: -5.1750336134513741048E+00              - 2.1271427013787699791E+00j
fpm: -5.17503361345137E+00                   - 2.12714270137877E+00j
apm: -5.1750336134513741059e+0 (-3.604e-14%) - 2.1271427013787699777e+0 (-7.287e-14%)j

Generalized Clausen cosine function

ctxflint.clausen_cos(s, z)

Returns the Clausen cosine function. See also Wikipedia [1465], MathWorld [1095], Mpmath [736].

Computes the Clausen cosine function, defined formally by the series

\[\mathrm{\widetilde{Cl}}_s(z) = \sum_{k=1}^{\infty} \frac{\cos(kz)}{k^s}.\]

This function is complementary to the Clausen sine function clsin(). In terms of the polylogarithm,

\[ \begin{align}\begin{aligned}\mathrm{\widetilde{Cl}}_s(z) = \frac{1}{2}\left(\mathrm{Li}_s\left(e^{iz}\right) + \mathrm{Li}_s\left(e^{-iz}\right)\right)\\= \mathrm{Re}\left[\mathrm{Li}_s(e^{iz})\right] \quad (s, z \in \mathbb{R}).\end{aligned}\end{align} \]

\(\quad\) \(\quad\)

3D wpf plot: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function \(z = \sin(x + iy)\), with \(-6 \le x \le 6\) (blue axis), \(-6 \le y \le 6\) (red axis), \(-10 \le z \le 10\) (green axis). Function values are loglog-transformed.

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; s = '3'; x = '4'
>>> \mathrm{d}x = dec.clcos(s, x); mx = mpm.clcos(s, x); gx = gmp.clcos(s, x)
>>> fx = fpm.clcos(s, x); ax = apm.clcos(s, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  -6.518926267198991308332758909517770104595E-1
mpm:  -6.518926267198991308332758909517770104595e-1
gmp:  -6.518926267198991308332758909517770104595E-01
fpm:  -6.51892626719899E-01
apm:  -6.518926267198991308332758909517763900500e-1 (-5.188e-29%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; s = '3'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.clcos(s, z); mz = mpm.clcos(s, z); gz = gmp.clcos(s, z)
>>> fz = fpm.clcos(s, z); az = apm.clcos(s, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: -2.1132834047743479436E+0               + 5.1271260828837504331E+0j
mpm: -2.1132834047743479436e+0               + 5.1271260828837504331e+0j
gmp: -2.1132834047743479436E+00              + 5.1271260828837504331E+00j
fpm: -2.11328340477435E+00                   + 5.12712608288375E+00j
apm: -2.1132834047743479423e+0 (-7.335e-14%) + 5.1271260828837504341e+0 (3.638e-14%)j

Classical Clausen function, \(\mathrm{Cl}_2(x)\)

math53.clausen2(x)

Returns the Clausen function \(\displaystyle \mathrm{Cl}_2(x) = \Im \displaystyle \mathrm{Li}_2(e^{ix}) = \int_0^x \log|2\sin(t/2)| \, \mathrm{d}t\).

See also: Wikipedia [1465], MathWorld [1095], Ehrhardt [309] (3.6.15), Mpmath [736].

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Clausen2(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Clausen2('0.1')
ereal('5.3518479027559984754E-1')

Bose-Einstein integrals, \(G_s(x)\)

math53.bose_einstein(s, x)

Returns the Bose-Einstein integral of real order \(s\), \(\displaystyle G_s(x) = \frac{1}{\Gamma(s+1)} \int_0^{\infty} \frac{t^s}{e^{t-x}-1} = \text{Li}_{s+1}(e^x) \,\). If \(x>0\) the real part of \(G_s(x)\) is returned.

See also: MathWorld [1024], Ehrhardt [309] (3.6.7), NIST [16].

\[G_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x}-1}\mathrm{d}t,\]

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.BoseEinstein(2,5)
ereal('5.2359877559829887307E-1')
>>> ereal.BoseEinstein(2,'51')
ereal('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; s = '10'; x = '0.5'
>>> \mathrm{d}x = dec.bose_einstein(s, x); mx = mpm.bose_einstein(s, x); gx = gmp.bose_einstein(s, x)
>>> fx = fpm.bose_einstein(s, x); ax = apm.bose_einstein(s, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  1.650075952950667128565820621131669931198E+0
mpm:  1.650075952950667128565820621131669931198e+0
gmp:  1.650075952950667128565820621131669931198E+00
fpm:  1.65007595295067E+00
apm:  1.650075952950667128565820642266282738366e+0 (2.139e-21%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; s = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.bose_einstein(s, z); mz = mpm.bose_einstein(s, z); gz = gmp.bose_einstein(s, z)
>>> fz = fpm.bose_einstein(s, z); az = apm.bose_einstein(s, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: -1.4096051587148306166E+2             + 1.9476656215692471738E+1j
mpm: -1.4096051587148306166e+2             + 1.9476656215692471738e+1j
gmp: -1.4096051587148306166E+02            + 1.9476656215692471738E+01j
fpm: -1.40960515871483E+02                 + 1.94766562156925E+01j
apm: -1.4096051581016388546e+2 (-7.08e-5%) + 1.9476656215507044363e+1 (0.0007137%)j

Fermi-Dirac integrals, \(F_s(x)\)

ctx.fermi_dirac(s, x)

where ctx is math53 or ctxflint.

Returns the Fermi-Dirac integral of order \(s\), \(\displaystyle F_s(x) = \frac{1}{\Gamma(s+1)} \int_0^{\infty} \frac{t^s}{e^{t-x}+1} = -\text{Li}_{s+1}(-e^x) \,\).

See also: Wikipedia [1425], MathWorld [1030], Ehrhardt [309] (3.6.8.1).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.FermiDiracR(2,5)
ereal('5.2359877559829887307E-1')
>>> ereal.FermiDiracR(2,'51')
ereal('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; s = '10'; x = '1.5'
>>> \mathrm{d}x = dec.fermi_dirac(s, x); mx = mpm.fermi_dirac(s, x); gx = gmp.fermi_dirac(s, x)
>>> fx = fpm.fermi_dirac(s, x); ax = apm.fermi_dirac(s, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  4.472317812068368662350774526763653861048E+0
mpm:  4.472317812068368662350774526763653861048e+0
gmp:  4.472317812068368662350774526763653861048E+00
fpm:  4.47231781206837E+00
apm:  4.472317812068368662350774520946064383782e+0 (1.711e-22%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; s = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.fermi_dirac(s, z); mz = mpm.fermi_dirac(s, z); gz = gmp.fermi_dirac(s, z)
>>> fz = fpm.fermi_dirac(s, z); az = apm.fermi_dirac(s, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: -1.5096075557263597965E+2              + 3.0204317063642472857E+1j
mpm: -1.5096075557263597965e+2              + 3.0204317063642472857e+1j
gmp: -1.5096075557263597965E+02             + 3.0204317063642472857E+01j
fpm: -1.50960755572636E+02                  + 3.02043170636425E+01j
apm: -1.5096075564796857312e+2 (-7.353e-5%) + 3.0204317086095309026e+1 (0.0004834%)j

Legendre’s Chi function, \(\chi_s(x)\)

math53.legendre_chi(s, x)

Returns Legendre’s Chi function, defined as \(\displaystyle \chi_s(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)^s} = \tfrac{1}{2} \left( \text{Li}_s(x) - \text{Li}_s(-x) \right) \,\), for \(s\ge 0, |x|\le 1\).

See also: Wikipedia [1434], MathWorld [1041], Ehrhardt [309] (3.6.9).

\(\quad\) \(\quad\)

3D wpf plot: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function \(z = \sin(x + iy)\), with \(-6 \le x \le 6\) (blue axis), \(-6 \le y \le 6\) (red axis), \(-10 \le z \le 10\) (green axis). Function values are loglog-transformed.

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.LegendreChi(2,5, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.LegendreChi('5.1', 0.5)
ereal('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; s = '10'; x = '1.5'
>>> \mathrm{d}x = dec.legendre_chi(s, x); mx = mpm.legendre_chi(s, x); gx = gmp.legendre_chi(s, x)
>>> fx = fpm.legendre_chi(s, x); ax = apm.legendre_chi(s, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  1.500058011828613644574663245595439676243E+0
mpm:  1.500058011828613644574663245595439676243e+0
gmp:  1.500058011828613644574663245595439676243E+00
fpm:  1.50005801182861E+00
apm:  1.500058011828613644574663246586459327124e+0 (2.88e-22%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; s = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.legendre_chi(s, z); mz = mpm.legendre_chi(s, z); gz = gmp.legendre_chi(s, z)
>>> fz = fpm.legendre_chi(s, z); az = apm.legendre_chi(s, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 4.9993028299912224901E+0             + 3.0032350570493713371E+0j
mpm: 4.9993028299912224901e+0             + 3.0032350570493713371e+0j
gmp: 4.9993028299912224901E+00            + 3.0032350570493713371E+00j
fpm: 4.99930282999122E+00                 + 3.00323505704937E+00j
apm: 4.9993028304470985305e+0 (2.364e-5%) + 3.0032350577905899996e+0 (4.62e-5%)j

Generalized inverse tangent integral

ctxflint.inverse_tan_integral(s, z)

Returns the generalized inverse tangent integral \(Ti_s(z)\), which can be expressed in terms of polylogarithms:

\[\mathrm {Ti} _{s}(z)={1 \over 2i}\left[\mathrm {Li} _{s}(iz)-\mathrm {Li} _{s}(-iz)\right].\]

The relation in particular implies:

\[\mathrm {Ti} _{0}(z)={z \over 1+z^{2}},\quad \mathrm {Ti} _{1}(z)=\arctan z,\quad \mathrm {Ti} _{2}(z)=\int _{0}^{z}{\arctan t \over t}\mathrm{d}t,\quad \ldots ~\quad \mathrm {Ti} _{n+1}(z)=\int _{0}^{z}{\frac {\mathrm {Ti} _{n}(t)}{t}}\mathrm{d}t,\]

which explains the function name.

See also: Wikipedia [1439], MathWorld [1037], Ehrhardt [309] (3.6.17).

\(\quad\) \(\quad\)

3D wpf plot: real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex sine function \(z = \sin(x + iy)\), with \(-6 \le x \le 6\) (blue axis), \(-6 \le y \le 6\) (red axis), \(-10 \le z \le 10\) (green axis). Function values are loglog-transformed.

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; s = '10'; x = '1.5'
>>> \mathrm{d}x = dec.ti(s, x); mx = mpm.ti(s, x); gx = gmp.ti(s, x)
>>> fx = fpm.ti(s, x); ax = apm.ti(s, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  1.499943569822004012305079650116043572538E+0
mpm:  1.499943569822004012305079650116043572538e+0
gmp:  1.499943569822004012305079650116043572538E+00
fpm:  1.49994356982200E+00
apm:  1.499943569822004012305079649905189176482e+0 (9.317e-24%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; s = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.ti(s, z); mz = mpm.ti(s, z); gz = gmp.ti(s, z)
>>> fz = fpm.ti(s, z); az = apm.ti(s, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 4.9998641623415218366E+0             + 2.9969552556663645569E+0j
mpm: 4.9998641623415218366e+0             + 2.9969552556663645569e+0j
gmp: 4.9998641623415218366E+00            + 2.9969552556663645569E+00j
fpm: 4.99986416234152E+00                 + 2.99695525566636E+00j
apm: 4.9998641636149685969e+0 (3.517e-5%) + 2.9969552549261524848e+0 (4.218e-5%)j