A linear fractional function ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'f(z)=rac{az+b}{cz+d}' 'utf-8' math_unknown_error where ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'a, b, c,' 'utf-8' math_unknown_error and ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'd' 'utf-8' math_unknown_error are complex constants with ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '{ad-bc} eq 0' 'utf-8' math_unknown_error has a pair of fixed points which are not necessarily distinct. This is so because the equation ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'z=rac{az+b}{cz+d}' 'utf-8' math_unknown_error is linear or quadratic in ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'z' 'utf-8' math_unknown_error. When the two fixed points coincide, the transformation is said to be parabolic.

When there are distinct fixed points, say ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'lpha' 'utf-8' math_unknown_error and ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'eta' 'utf-8' math_unknown_error, then we can write the transformation in the form ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'rac{w-lpha}{w-eta}=rac{z-lpha}{z-eta}' 'utf-8' math_unknown_error. When ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'k' 'utf-8' math_unknown_error is real and necessarily non-zero the transformation is said to be hyperbolic. When ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|k| = 1' 'utf-8' math_unknown_error, the transformation is said to be elliptic.