This is called the exponential map, and it maps the Lie algebra formula_93 into the Lie group formula_2. It provides a diffeomorphism between a neighborhood of 0 in formula_93 and a neighborhood of formula_141 in formula_2. This exponential map is a generalization of the exponential function for real numbers (because formula_6 is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because formula_7 is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because formula_145 with the regular commutator is the Lie algebra of the Lie group formula_146 of all invertible matrices).