Polylogarithm

Polylogarithm. The polylogarithm is a special function of order $s$ and argument $z$, central to complex analysis and number theory. It generalizes the natural logarithm and the Riemann zeta function, appearing in solutions of Feynman integrals within quantum electrodynamics and the study of Fermi–Dirac statistics. Its properties are deeply intertwined with those of other special functions like the Hurwitz zeta function and Lerch transcendent.

Definition and basic properties

The classical definition of the polylogarithm is the infinite series $\backslash operatorname\{Li\}\_s(z)\; =\; \backslash sum\_\{k=1\}^\backslash infty\; \{z^k\}/\{k^s\}$, which converges for and any complex order $s$. This definition can be extended to the entire complex $z$-plane via analytic continuation, except for a branch point at $z=1$. For integer orders, it satisfies a differential-difference equation related to the work of Augustin-Louis Cauchy. The function is multivalued, and its principal branch is typically defined by a branch cut along the real axis from $1$ to $+\backslash infty$. Key functional equations, such as the inversion formula, connect $\backslash operatorname\{Li\}\_s(z)$ to $\backslash operatorname\{Li\}\_s(1/z)$, revealing symmetries studied by Leonhard Euler.

Special cases and values

Important special cases arise for specific values of the order $s$. For $s=1$, the polylogarithm reduces to the dilogarithm, also known as Spence's function, which solves certain integral equations. The case $s=2$ gives the trilogarithm, which appears in calculations of higher-loop corrections in quantum chromodynamics. When $s$ is a negative integer, the polylogarithm becomes a rational function, expressible via the Eulerian numbers. At $z=1$, the function evaluates to the Riemann zeta function, $\backslash zeta(s)$, a cornerstone of the Riemann hypothesis. Other notable values include connections to the Dirichlet beta function and Catalan's constant at specific arguments.

Integral representations and identities

The polylogarithm admits several integral representations, crucial for its analytic continuation and evaluation. One fundamental representation is $\backslash operatorname\{Li\}\_s(z)\; =\; \{z\}/\{\backslash Gamma(s)\}\; \backslash int\_0^\backslash infty\; \{t^\{s-1\}\}/(\{e^t\; -\; z\})\backslash ,\; dt$ for $\backslash Re(s)\; >\; 0$, which involves the gamma function. Another, derived via Hankel contour integration, is valid for all complex $s$. These representations lead to important identities, such as Jonquière's identity, which relates polylogarithms of different orders. The function also satisfies a duplication formula analogous to that of the Legendre chi function. Relationships with the incomplete gamma function and the Lerch zeta function are established through these integrals.

Series representations and asymptotics

Beyond its defining power series, the polylogarithm possesses useful expansions near its singularities. An expansion for $|z|\; \backslash ll\; 1$ is given by the defining series itself. Near the branch point $z=1$, an asymptotic expansion involves the Riemann zeta function and powers of $\backslash ln(-\backslash ln\; z)$, relevant in perturbation theory. For large $|z|$, an expansion in inverse powers of $\backslash ln(-z)$ is employed, connected to work by Niels Henrik Abel. The Bernoulli numbers appear in series for polylogarithms of non-positive integer order. These asymptotic forms are vital in evaluating Feynman diagrams and in statistical mechanics for systems obeying Bose–Einstein statistics.

Applications

The polylogarithm is indispensable in theoretical physics and mathematics. In statistical mechanics, it describes the distribution of ideal quantum gases, with the Fermi–Dirac integral and Bose–Einstein integral expressible in its terms. Within quantum field theory, it emerges from the computation of multi-loop integrals, particularly in precision tests of the Standard Model. In number theory, it appears in the study of special values of L-functions and Mahler measures. The function also arises in analytic number theory through connections to the prime number theorem and in the analysis of algorithmic complexity via the Lambert W function. Its utility extends to electromagnetic theory in calculating the internal inductance of conductors. Category:Special functions Category:Mathematical physics Category:Complex analysis