Lobachevsky function

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.

definition

The Lobachevsky function is defined as an integral

${\displaystyle \Lambda (\theta )=-\int _{0}^{\theta }\log \mid 2\sin u\mid du}$

properties

The Lobachevsky function is continuous , periodic with a period and an odd function . It has a uniformly converging Fourier series ${\displaystyle \pi }$

${\displaystyle \Lambda (\theta )={\frac {1}{2}}\sum _{n=1}^{\infty }\sin(2n\theta )/n^{2}}$.

Your derivatives are

${\displaystyle \Lambda ^{\prime }(\theta )=-\log \mid 2\sin \theta \mid ,\Lambda ^{\prime \prime }(\theta )=-\cot \theta }$.

You have the functional equation

${\displaystyle \Lambda (n\theta )=\sum _{j\ mod\ n}n\Lambda (\theta +j\pi /n)}$

for all integers . ${\displaystyle n\not =0}$

for true ${\displaystyle 0<\theta <\pi }$

${\displaystyle Li_{2}(e^{i\theta })=Li_{2}(1)+\theta (\pi -\theta )+2i\Lambda (\theta )}$,

${\displaystyle 2\Lambda (\theta )}$is therefore the imaginary part of the (classical) dilogarithm of . The relationship with the Bloch-Wigner dilogarithm results from the equation ${\displaystyle e^{i\theta }}$

${\displaystyle D(e^{i\theta })=2\Lambda ({\frac {\theta }{2}})}$.

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: ${\displaystyle \alpha ,\beta ,\gamma }$ ${\displaystyle \alpha +\beta +\gamma =\pi }$ ${\displaystyle T_{\alpha ,\beta ,\gamma }}$

${\displaystyle Vol(T_{\alpha ,\beta ,\gamma })=\Lambda (\alpha )+\Lambda (\beta )+\Lambda (\gamma )}$.

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.