Complex-valued specular differentiation

Develop specular differentiation over the complex numbers in order to extend the stability analysis of the specular ellipse method beyond real negative coefficients.

Background

The paper analyzes the stability of the specular ellipse method for the real Dahlquist test equation with a negative coefficient. The method’s scaled angular mean is not homogeneous, and the authors establish unconditional decay for arbitrary positive scale sequences when the coefficient is real and negative. They explicitly limit this analysis because the underlying specular differentiation framework has not been developed over the complex numbers; extending that framework would be necessary to study complex coefficients or a broader complex stability theory.