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== Angle of parallelism ==

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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l060020a.gif" />

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.

== Second meaning ==

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.

==Comments==
For the Lobachevskii function in the sense of 1) (i.e. the angle of parallelism) see also [[#References|[a1]]]–[[#References|[a4]]].

For Lobachevskii's function as defined in 2) see also [[#References|[a5]]].

==References==
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,   "Foundations of geometry" , '''1''' , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.V. Efimov,   "Höhere Geometrie" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Ryzhik,   I.S. Gradshtein,   "Tables of integrals, series, and products" , Acad. Press  (1980)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Greenberg,   "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,   "Parallel lines"  ''Canad. Math. Bull.'' , '''21'''  (1978)  pp. 385–397</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,   "Non-Euclidean geometry" , Univ. Toronto Press  (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Bonola,   "Non-Euclidean geometry" , Dover, reprint  (1955)  (Translated from Italian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.S.M. Coxeter,   "Twelve geometric esays" , Carbondale  (1968)  pp. Chapt. 1</TD></TR></table>

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