how to prove this equation?

Question

Let f and g be the entire functions. If |z| = 1, then f(z)=g(z). Prove that f(z)=g(z) for all z belong to complex.

Answer 1

I think you mean "Let f and g be entire functions," which implies that f and g are analytic functions.

Since f and g are analytic, then f-g is also analytic. However, for all |z|=1,

f(z)-g(z) = 0

To quote Wikipedia, "If the set of zeros of an analytic function f has an accumulation point inside its domain, then f is zero everywhere on the connected component containing the accumulation point.

More formally this can be stated as follows. If (rn) is a sequence of distinct numbers such that f(rn) = 0 for all n and this sequence converges to a point r in the domain of D, then f is identically zero on the connected component of D containing r."

Thus, f(z)-g(z) must be identically zero for all complex z. In that case, f(z)=g(z) for all complex z.