Extension of the Multi-Valued Radical Representation to Quintics with Complex Coefficients: Branch Cut Analysis, Riemann Surface, and Complete Algorithm

We extend the multi-valued radical representation previously developed for real Brioschi quintics to the fully complex case. For any complex parameter C \in \mathbb{C}, the Brioschi normal form y^{5} - 5y^{3} + 5y = C is solved by the same radical expression: y = \sqrt[5]{\frac{C}{2} + \sqrt{\left(\frac{C}{2}\right)^{2} - 1}} + \sqrt[5]{\frac{C}{2} - \sqrt{\left(\frac{C}{2}\right)^{2} - 1}}. However, the branch cuts of the square root and the fifth root now require a thorough analysis in the complex plane. We identify branch points at C = \pm 2 and at infinity, define principal branches consistent with the real case, and provide a complete algorithm to generate all five roots for any complex C. We also study the Riemann surface of the function y(C) and its monodromy around the branch points. Numerical examples with complex coefficients are given, including full tables of five roots for various complex C. Classical reductions (Tschirnhaus, Bring-Jerrard, Brioschi-Klein) are extended to complex coefficients, enabling the solution of any complex quintic. This work complements our previous real-case paper and opens the door to applications in dissipative systems, PT-symmetric quantum mechanics, and complex analysis. Keywords: Complex quintic equations, Brioschi normal form, multi-valued radical representation, branch cut, Riemann surface, complex Tschirnhaus transformation, monodromy, PT-symmetry.