dilogarithmifunktio

The dilogarithm function, often denoted as Li₂(z), is a special function in mathematics. It is a polylogarithm of order 2. The dilogarithm can be defined by its Maclaurin series expansion for |z| < 1 as the sum of zⁿ/n² from n=1 to infinity. This series converges absolutely for |z| ≤ 1, except at z=1.

Another way to define the dilogarithm is through an integral representation. For any complex number z, it can be expressed as the integral of -ln(1-t)/t dt from 0 to z. This integral representation is valid for all complex numbers z except for real numbers z > 1.

The dilogarithm function possesses several important properties. It satisfies various functional equations, some of which are quite intricate. For instance, the Landen's identity and the inversion formula are notable examples. The dilogarithm also appears in various areas of physics, particularly in quantum field theory and statistical mechanics, where it often arises in the evaluation of Feynman diagrams and in the study of critical phenomena. Its connection to other special functions, like the Gamma function and the Riemann zeta function, further highlights its significance in mathematical analysis. The dilogarithm has a branch cut along the real axis from 1 to infinity.