L'Hopital rules for complex-valued functions in higher dimensions

In calculus, l’Hopital’s rule provides a simple way to evaluate the limits of quotient functions when both the numerator and denominator vanish. But what happens when we move beyond real functions on a real interval? In this article, we study when the quotient of two complex-valued functions in higher dimension can be defined continuously at the points where both functions vanish. Surprisingly, the answer is far subtler than in the real-valued setting. We provide a complete characterization for the continuity of the quotient function. We also point out why extending this result to smoother quotients remains an intriguing challenge.

💡 Research Summary

The paper investigates how the classical L’Hôpital rule, which gives a simple method for evaluating limits of quotients when both numerator and denominator vanish, can be extended to complex‑valued functions defined on higher‑dimensional real domains. After recalling the well‑known real‑valued one‑dimensional and multivariate versions (Theorems 1 and 2), the authors turn to the complex case and discover that the situation is dramatically more subtle.

A key notion introduced is that of a simple zero for a complex‑valued function f : Ω⊂ℝⁿ→ℂ: at each zero a the differential Df|ₐ, viewed as a real linear map ℝⁿ→ℝ², must have full rank 2. Consequently the zero set Γ_f is a smooth submanifold of codimension 2. When two smooth complex functions f and g share the same simple zero set Γ, their differentials at any a∈Γ are both surjective, and there exists a unique 2×2 real matrix Aₐ such that

 Df|ₐ = Aₐ · Dg|ₐ.

This matrix is invariant under smooth changes of coordinates in the domain, so it encodes an intrinsic relationship between the two functions at the common zero.

The authors first show (Theorem 3) that for any smooth path γ crossing Γ transversally (i.e., g′_γ ≠ 0), the directional limit of the quotient f/g along γ exists and equals

 lim_{t→0} f(γ(t))/g(γ(t)) = (1/g′γ) · A{γ(0)} · g′_γ,

where g′γ is the complex number Dg|{γ(0)}(γ′(0)). This “path‑dependent L’Hôpital rule” explains why in Examples 1 and 2 the limit depends on the slope of the line approaching the origin.

The central result (Theorem 4) characterises when a continuous quotient φ exists such that f = g φ. The necessary and sufficient condition is that the matrices Aₐ be scaled rotations at every a∈Γ, i.e.

 Aₐ =