# Lobachevskii function

The angle of parallelism in Lobachevskii geometry is a function that expresses the angle $\alpha$ between the line $u _ {1}$( or $u _ {2}$) (see Fig.) and the segment $OA$ perpendicular to a line $a$ parallel to $u _ {1}$( or $u _ {2}$) in terms of the length $l$ of the segment $OA$:

$$\alpha = \Pi ( l) = 2 { \mathop{\rm arc} \mathop{\rm tan} } e ^ {- l / R } ,$$

where $R$ is a positive constant that corresponds to the scale of measurement of distances.

Figure: l060020a

The Lobachevskii function is a continuous monotone decreasing function with values between $\pi /2$ and 0:

$$\lim\limits _ {l \rightarrow 0 } \Pi ( l) = \frac \pi {2} ,\ \lim\limits _ { l \rightarrow \infty } \Pi ( l) = 0.$$

It was introduced by N.I. Lobachevskii in 1826.

## Contents

#### References

 [1] V.F. Kagan, "Foundations of geometry" , 1 , Moscow-Leningrad (1949) (In Russian) [2] N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)

The special function (cf. Special functions) defined for real $x$ by

$$L ( x) = - \int\limits _ { 0 } ^ { x } \mathop{\rm ln} \cos t dt .$$

The Lobachevskii function can be represented as a series

$$L ( x) = x \mathop{\rm ln} 2 - \frac{1}{2} \sum _ { k= } 1 ^ \infty (- 1) ^ {k-} 1 \frac{\sin 2kx }{k ^ {2} } .$$

The main relations are:

$$L ( - x ) = - L ( x) ,\ - \frac \pi {2} \leq x \leq \frac \pi {2} ,$$

$$L ( \pi - x ) = \pi \mathop{\rm ln} 2 - L ( x) ,$$

$$L ( \pi + x ) = \pi \mathop{\rm ln} 2 + L ( x) .$$

It was introduced by N.I. Lobachevskii in 1829.

#### References

 [1] I.M. Ryzhik, I.S. Gradshtein, "Tables of integrals, series, and products" , Acad. Press (1980) (Translated from Russian)