Asian Journal of Pure and Applied Mathematics

This work establishes a nonlocal analytic framework in the complex plane by introducing a symmetric fractional complex derivative constructed through symmetric complex increments. The proposed operator differs from classical Riemann—Liouville and Caputo derivatives and preserves rotational compatibility and phase–consistent scaling in the complex plane. Using this framework, we derive fractional analogues of Cauchy conditions, a nonlocal Cauchy—Pompeiu identity, and a fractional Liouville theorem, together with a Laurent-–Mittag-–Leffler type expansion. The analysis is developed using symmetric increment techniques and fractional contour integral representations. As a proof of concept, a fractional harmonic potential model is examined, demonstrating memory-driven deformation in analytic flows. The framework provides a mathematical foundation for fractional holomorphic geometry, memory influenced signal processing, and nonlocal complex PDE models.