Lobachevskii function

The angle of parallelism in Lobachevskii geometry is a function that expresses the angle between the line (or ) (see Fig.) and the segment perpendicular to a line parallel to (or ) in terms of the length of the segment :

where 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 and 0:

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

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 by

The Lobachevskii function can be represented as a series

The main relations are:

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)

Comments

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

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

References

[a1]	M. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974)
[a2]	H.S.M. Coxeter, "Parallel lines" Canad. Math. Bull. , 21 (1978) pp. 385–397
[a3]	H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1957)
[a4]	R. Bonola, "Non-Euclidean geometry" , Dover, reprint (1955) (Translated from Italian)
[a5]	H.S.M. Coxeter, "Twelve geometric esays" , Carbondale (1968) pp. Chapt. 1