Proposition
Erdös contends that in Complex Analysis, versus Single Variable Calculus, a function will not have a derivative of zero when it reaches a maximum value.
In Single Variable Calculus:
A function f(x), which is smooth (continuous and differentiable) on a open domain D, will have a derivative of zero when it attains a maximum on D.
However, for a complex function g(z), on a closed domain U with border B, the maximum (Guaranteed by the Extreme Value Theorem), may not have a derivative of zero.
Proof
Let there exist a complex ϵ such that |ϵ|>0 and δ such that |δ|>0 where ϵ and δ are arbitrarily small.
Furthermore, let the complex function f(z) be analytic and non-constant on an open domain D, bounded by a contour B along which f(z) is continuous and differentiable. Thus, f is continuous and differentiable on the closed region U=D∪B.
By the Maximum Modulus Theorem, the maximum of f occurs on B at z0.
Now, suppose that f′(z0) is indeed zero. This would mean that for some ϵ, there is are points w=z0+δ where δ<ϵ for which f(w)=f(z0), since f(z) is continuous. z0+δ is a maximum point for f(z) on D. By the Maximum Modulus Theorem, this means that f must be constant on D.
Since f(z) is non-constant on D, this is a contradiction. #
Furthermore, let the complex function f(z) be analytic and non-constant on an open domain D, bounded by a contour B along which f(z) is continuous and differentiable. Thus, f is continuous and differentiable on the closed region U=D∪B.
By the Maximum Modulus Theorem, the maximum of f occurs on B at z0.
Now, suppose that f′(z0) is indeed zero. This would mean that for some ϵ, there is are points w=z0+δ where δ<ϵ for which f(w)=f(z0), since f(z) is continuous. z0+δ is a maximum point for f(z) on D. By the Maximum Modulus Theorem, this means that f must be constant on D.
Since f(z) is non-constant on D, this is a contradiction. #