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|newpage| Polylogarithm, and related functions =============================================================================== Polylogarithm of integer order, `\mathrm{Li}_n(x)` ------------------------------------------------------------------------------- .. method:: math53.polylog_i(n, x) Returns the polylogarithm of integer order `n`, `\displaystyle \mathrm{Li}_n(x) = \sum_{k=1}^{\infty} \frac{x^k}{k^n}, \quad n \in \mathbb{Z}, |x|<1 \,`, or its analytic continuation; for `n>0`, `x>1` the real part of `\mathrm{Li}_n(x)` is returned. See also Wikipedia :cite:p:`WikipediaFun173`, MathWorld :cite:p:`WolframFun173`, NIST :cite:p:`DLMFun173`, :cite:t:`Ehrhardt2018` (3.6.11). Returns the polylogarithm function of integer order `n`. This function returns the polylogarithm function of integer order `n` .. math :: \text{Li}_n(z)=\sum_{k=1}^\infty \frac{z^{k}}{k^n}, \quad n \in \mathbb{Z}, |z|<1. or its analytic continuation; for `n>0` there is the arguments restriction `x\leq 1`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Polylog(2, 0.5) xreal('5.2359877559829887307E-1') >>> xreal.Polylog(2, '0.1') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Polylog(2, 0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Polylog(2, '0.1') Gpr('5.3518479027559984754E-1') Fermi-Dirac integrals of integer order, `F_n(x)` ------------------------------------------------------------------------------- .. method:: math53.fermi_dirac_i(n, x) Returns the Fermi-Dirac integral of integer order `n`, `\displaystyle F_s(x) = \frac{1}{\Gamma(n+1)} \int_0^{\infty} \frac{t^n}{e^{t-x}+1} = -\text{Li}_{n+1}(-e^x) \,`. See also: Wikipedia :cite:p:`WikipediaFun173c`, MathWorld :cite:p:`WolframFun173c`, :cite:t:`Ehrhardt2018` (3.6.8.2). This function returns the complete Fermi-Dirac integrals `F_n(x)` of integer order. They are defined for real orders `s>-1` by .. math :: F_s(x)=\frac{1}{\Gamma(s+1)} \int_0^\infty \frac{t^s}{e^{t-x}+1} \mathrm{d}t and by analytic continuation for `s \leq -1` using polylogarithms .. math :: F_s(x)=-\text{Li}_{s+1}(-e^x). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.FermiDirac(2,5) xreal('5.2359877559829887307E-1') >>> xreal.FermiDirac(2,'51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.FermiDirac(2,5) Gpr('5.2359877559829887307E-1') >>> Gpr.FermiDirac(2,'51') Gpr('5.3518479027559984754E-1') Fermi-Dirac integral `F_{-1/2}(x)` ------------------------------------------------------------------------------- .. method:: math53.fermi_dirac_m05(x) Returns the Fermi-Dirac integral `F_{-1/2}(x)`. See also: Wikipedia :cite:p:`WikipediaFun173c`, MathWorld :cite:p:`WolframFun173c`, :cite:t:`Ehrhardt2018` (3.6.8.3). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.FermiDiracm05(2,5) xreal('5.2359877559829887307E-1') >>> xreal.FermiDiracm05('5.1') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.FermiDiracm05(2,5) Gpr('5.2359877559829887307E-1') >>> Gpr.FermiDiracm05('5.1') Gpr('5.3518479027559984754E-1') Fermi-Dirac integral `F_{1/2}(x)` ------------------------------------------------------------------------------- .. method:: math53.fermi_dirac_p05(x) Returns the Fermi-Dirac integral `F_{1/2}(x)`. See also: Wikipedia :cite:p:`WikipediaFun173c`, MathWorld :cite:p:`WolframFun173c`, :cite:t:`Ehrhardt2018` (3.6.8.3). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.FermiDiracp05(2,5) xreal('5.2359877559829887307E-1') >>> xreal.FermiDiracp05('5.1') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.FermiDiracp05(2,5) Gpr('5.2359877559829887307E-1') >>> Gpr.FermiDiracp05('5.1') Gpr('5.3518479027559984754E-1') Fermi-Dirac integral `F_{3/2}(x)` ------------------------------------------------------------------------------- .. method:: math53.fermi_dirac_p15(x) Returns the Fermi-Dirac integral `F_{3/2}(x)`. See also: Wikipedia :cite:p:`WikipediaFun173c`, MathWorld :cite:p:`WolframFun173c`, :cite:t:`Ehrhardt2018` (3.6.8.3). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.FermiDiracp15(2,5) xreal('5.2359877559829887307E-1') >>> xreal.FermiDiracp15('5.1') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.FermiDiracp15(2,5) Gpr('5.2359877559829887307E-1') >>> Gpr.FermiDiracp15('5.1') Gpr('5.3518479027559984754E-1') Fermi-Dirac integral `F_{5/2}(x)` ------------------------------------------------------------------------------- .. method:: math53.fermi_dirac_p25(x, s) Returns the Fermi-Dirac integral `F_{5/2}(x)`. See also: Wikipedia :cite:p:`WikipediaFun173c`, MathWorld :cite:p:`WolframFun173c`, :cite:t:`Ehrhardt2018` (3.6.8.3). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.FermiDiracp25(2,5) xreal('5.2359877559829887307E-1') >>> xreal.FermiDiracp25('5.1') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.FermiDiracp25(2,5) Gpr('5.2359877559829887307E-1') >>> Gpr.FermiDiracp25('5.1') Gpr('5.3518479027559984754E-1') Inverse tangent integral, `\mathrm{Ti}_2(x)` ------------------------------------------------------------------------------- .. method:: math53.tangent_int_2(x) Returns the inverse tangent integral `\displaystyle \mathrm{Ti}_2(x) = \int_0^x \frac{\arctan(t)}{t} \, \mathrm{d}t`. See also: Wikipedia :cite:p:`WikipediaFun173a`, MathWorld :cite:p:`WolframFun173a`, :cite:t:`Ehrhardt2018` (3.6.16). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.TangentInt2(0.5) xreal('5.2359877559829887307E-1') >>> xreal.TangentInt2('0.1') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.TangentInt2(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.TangentInt2('0.1') Gpr('5.3518479027559984754E-1') Lobachevsky's log-cos integral, `L(x)` ------------------------------------------------------------------------------- .. method:: math53.lobachevsky_c(x) Returns Lobachevsky's log-cos integral `\displaystyle L(x) = \int_0^x \log|\cos(t)| \, \mathrm{d}t`. See also: :cite:t:`Ehrhardt2018` (3.6.18). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.LobachevskyC(0.5) xreal('5.2359877559829887307E-1') >>> xreal.LobachevskyC('0.1') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.LobachevskyC(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.LobachevskyC('0.1') Gpr('5.3518479027559984754E-1') Lobachevsky's log-sin integral, `\Lambda(x)` ------------------------------------------------------------------------------- .. method:: math53.lobachevsky_s(x, s) Returns Lobachevsky's log-sin integral `\displaystyle \Lambda(x) = \int_0^x \log|2\sin(t)| \, \mathrm{d}t`. See also: :cite:t:`Ehrhardt2018` (3.6.19). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.LobachevskyS(0.5) xreal('5.2359877559829887307E-1') >>> xreal.LobachevskyS('0.1') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.LobachevskyS(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.LobachevskyS('0.1') Gpr('5.3518479027559984754E-1') Debye functions, `\mathrm{D}_n(x)` ------------------------------------------------------------------------------- .. method:: math53.debye(n,x) Returns the Debye function `\displaystyle \mathrm{D}_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^t-1} \, \mathrm{d}t, \quad n>0, x \ge 0`. See also: Wikipedia :cite:p:`WikipediaFun323`, MathWorld :cite:p:`WolframFun323`, :cite:t:`Abramowitz1970` section 27.1, :cite:t:`Ehrhardt2018` (3.10.6), :cite:t:`Dubinov2008`. This routine returns the Debye functions .. math:: \text{D}_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^t-1} \mathrm{d}t \quad (n>0, x \geq 0). .. math:: \text{D}_n^{(1)}(x) = \int_0^x \frac{t^n}{e^t-1} \mathrm{d}t = n! \zeta(n+1) - n! Z_{n+1}(z) . .. math:: \text{D}_n^{(2)}(x) = \int_x^{\infty} \frac{t^n}{e^t-1} \mathrm{d}t = n! Z_{n+1}(z). .. math:: \text{D}^{(2)}_n(x) = \Gamma(n+1)\zeta(n+1) - \text{D}^{(1)}_n(x). .. math:: \text{D}^{(1)}_n(x) = n! \zeta(n+1) - \text{D}^{(2)}_n(x). .. math:: Z_{n+1}(z)=\sum _{k=0}^{n}\mathrm{Li} _{n-k+1}(e^{-z}){z^{k} \over k!} \quad (n=1,2,3,\ldots ). In terms of the incomplete zeta functions or "Debye functions" (Abramowitz & Stegun 1972, § 27.1): .. math:: Z_{n}(z)={1 \over (n-1)!}\int _{z}^{\infty }{t^{n-1} \over e^{t}-1}\mathrm{d}t\qquad (n=1,2,3,\ldots ), the polylogarithm Lin(z) for positive integer n may be expressed as the finite sum (Wood 1992, § 16): .. math:: \mathrm {Li} _{n}(e^{\mu })=\sum _{k=0}^{n-1}Z_{n-k}(-\mu ){\mu ^{k} \over k!}\qquad (n=1,2,3,\ldots ). A remarkably similar expression relates the "Debye functions" Zn(z) to the polylogarithm: .. math:: Z_{n}(z)=\sum _{k=0}^{n-1}\mathrm{Li}_{n-k}(e^{-z}){z^{k} \over k!}\qquad (n=1,2,3,\ldots ). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Debye(1,3) xreal('5.2359877559829887307E-1') >>> xreal.Debye(2,13) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Debye(1,3) Gpr('5.2359877559829887307E-1') >>> Gpr.Debye(2,13) Gpr('5.3518479027559984754E-1') Transport integral, `\mathrm{J}_n(x)` ------------------------------------------------------------------------------- .. method:: math53.transport_jn(n,x) Returns the transport function `\displaystyle \mathrm{J}_n(x) =\int_0^x \frac{t^n e^t}{(e^t-1)^2} \, \mathrm{d}t, \quad n \ge 2, x \ge 0`. See also: :cite:t:`Ehrhardt2018` (3.10.24). A.J. MacLeod, The numerical computation of transport integrals, 1992, Computer Physics Communications, 69, pp. 229-234, https://doi.org/10.1016/0010-4655(92)90162-R, An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.TransportJn(2,3) xreal('5.2359877559829887307E-1') >>> xreal.TransportJn(4,13) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.TransportJn(2,3) Gpr('5.2359877559829887307E-1') >>> Gpr.TransportJn(4,13) Gpr('5.3518479027559984754E-1')