In this paper a new analytic method is given, in which the quantum gro
up theory and the classical analysis are fused into a noncommutative a
nalysis. In this noncommutative analysis the Maurer-Cartan equation on
the quantum plane can be solved. The classical Riemann-Hilbert proble
m can be extended to the noncommutative analysis, and by using this Ri
emann-Hilbert problem, some nontrivial new solutions of the quantum Ma
urer-Cartan equation can be generated.