Abstract: The approximation of analyticfunctions with singularity f-(x)=1(x-a)/2 and g(x)=ln(1+x) are investigated using Chebyshev polynomials.Furthermore,their exponential approximation degrees are given.The approximation of the derivative for the best polynomial of f(x)=(x-a)-1 to f′(x) is also studied.The results suggest that approximation based on the Chebyshev polynomials has excellent effect on some functions.

Key words: Chebyshev polynomial, convergence rate, simultaneous approximation