We develop a calculus on open subsets of ℂⁿ intermediate between real differentiability and complex analyticity. Given a smooth nonvanishing field v: U→ℂⁿ\0, a function φ: U→ℂ is called v-differentiable if dφ (v) (z) ∈ ℝ for all z ∈ U. The v-derivative ∂ᵥ φ (z): = dφ (v) (z) ∈ ℝ produces a real number from a complex function in any dimension. We establish: (i) algebraic structure of Dᵥ (U) ; (ii) a fundamental theorem of integration recovering only the real part, with imaginary part frozen as topological invariant Ω_γ; (iii) classification of fields v by integrability, normalization, and residue structure; (iv) v-Taylor series with real coefficients and explicit radius of convergence; (v) v-differential equations reducing via w = Φ (z) to real first-order ODEs parametrized by Ω_γ, unifying the Aₙ hierarchy as solutions of ∂ₕ䂸φ = -1; (vi) a complete family of v-elementary functions; and (vii) extension to ℂⁿ, revealing a dimensional reduction theorem. The maximal bifurcation theorem — that Ω_γ is invisible to every operation of the v-calculus — holds in all dimensions.
Judicael Brindel (Mon,) studied this question.