The Lobachevski function (ger .: Lobachevsky function ) is closely related to the Clausen function and the Dilogarithm related special function of mathematics . The name goes back to John Milnor , Lobachevsky had used similar functions to calculate hyperbolic volumes.
is therefore the imaginary part of the (classical) dilogarithm of . The relationship with the Bloch-Wigner dilogarithm results from the equation
.
Volume of hyperbolic simplices
An ideal simplex in 3-dimensional hyperbolic space is determined by its six edge angles, with opposing angles being equal and the three non-opposing angles satisfying the equation . The volume of the ideal simplex can be calculated using the Lobachevsky function:
.
With the help of this formula there are numerous other formulas for volumes of hyperbolic polyhedra that use the Lobachevsky function.
literature
John Milnor: Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (NS) 6 (1982) no. 1, 9-24. on-line
JG Ratcliffe, Foundations of hyperbolic manifolds (2nd ed.). Graduate Texts in Mathematics 149, Springer, 2006.