The quantum logarithm is a function of mathematical physics .
definition
Be it . The quantum logarithm


is defined by
-
,
where is a curve running along the real axis from to and revolving around the zero point from above, for example .



![{\ displaystyle C = \ left [- \ infty, -1 \ right] \ cup \ left \ {e ^ {it} \ colon \ pi \ geq t \ geq 0 \ right \} \ cup \ left [1, \ infty \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51b2463f547888ada2854aba4f54782184a6b063)
(For every curve with these properties, integration of this integrand over the curve gives the same value.)
properties









The 1-form is meromorphic , it has simple poles with a residual in the points with .



literature
- VV Fock, AB Goncharov: The quantum dilogarithm and representations of quantum cluster varieties. Invent. Math. 175 (2009), no. 2, 223-286. (Chapter 4.1)