In this research westudied singular Cauchy’s integral for functionsbelong to wide
classes of functions on a famous curves families .Especially we study the boundness of this
integral. Wehave obtained some results about singular Cauchy’s integral and it’sboundness
for some functional classes branched from Lebesuge classes.
In this research, we have studied the issue of approximation of complex functions from weighted Lebesgue space ; and (Mukenhoupt weight) to rational functions by using p- Faber polynomials on large group of curves, which called Carlson curves. This
is also considered as a follow-up to the work done by researchers: Israfilov and Testici in 2014 , where they studied approximation of functions from weighted Smirnov space on domains with a Carlson curve boundary.
We study in this research approximation of complex functions from Orlicz space on a subclass of Carlson curves, which called Dini smooth curves to rational functions by using polynomials related with Dzjadyk sums which obtained from Faber polynomials. We depend on some concepts of complex analysis such as formulas of Sokhotski to reach the desired goal
