Problem 1
Use the precise definitions, state all hypotheses, apply f'(z_0)=lim_{h→0}(f(z_0+h)-f(z_0))/h, and verify the conclusion against a boundary case or independent theorem.
Faculty solution guide
Complex Differentiability and Holomorphic Functions
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Complex Differentiability and Holomorphic Functions.
Problem 4
The argument must address requiring one derivative independent of approach direction with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in fluid flow, electromagnetism, signal processing, geometry, and dynamical systems, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.