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

Albert Chern, Sadashige Ishida

TL;DR

This paper studies when the quotient $f/g$ of two smooth complex-valued functions on an open set $\Omega\subset\mathbb{R}^n$ extends continuously across a common simple zero set $\Gamma$. It shows that, unlike the real case, simple zeros alone do not guarantee a smooth quotient, and develops a framework based on the derivative-ratio $Df/Dg$ to characterize continuity. A central result proves that a continuous quotient exists if and only if $f$ and $g$ are complex linearly related, i.e., $(Df/Dg)_{\mathbf{a}}$ acts as a scaled rotation on $\mathbb{R}^2$ for all $\mathbf{a}\in\Gamma$, with a nuanced discussion of how this reduces to path-independent limits in 2D but not necessarily in higher dimensions. The paper also outlines open problems for achieving higher regularity of the quotient and discusses generalizations to coordinate changes, manifolds, and lower-regularity settings.

Abstract

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.

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

TL;DR

This paper studies when the quotient of two smooth complex-valued functions on an open set extends continuously across a common simple zero set . It shows that, unlike the real case, simple zeros alone do not guarantee a smooth quotient, and develops a framework based on the derivative-ratio to characterize continuity. A central result proves that a continuous quotient exists if and only if and are complex linearly related, i.e., acts as a scaled rotation on for all , with a nuanced discussion of how this reduces to path-independent limits in 2D but not necessarily in higher dimensions. The paper also outlines open problems for achieving higher regularity of the quotient and discusses generalizations to coordinate changes, manifolds, and lower-regularity settings.

Abstract

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.
Paper Structure (11 sections, 9 theorems, 17 equations, 3 figures)

This paper contains 11 sections, 9 theorems, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quotient of Complex Functions
  3. Ratio of derivatives
  4. Continuous Quotient of Complex Multivariable Functions
  5. Open problem
  6. Generalizations
  7. First Order Taylor Expansion
  8. Proof of Theorem \ref{['thm:RealQuotient1D']}
  9. Proof of Theorem \ref{['thm:RealQuotientnD']}
  10. Proof of Theorem \ref{['thm:PathDependentlHopital']}
  11. Proof of Theorem \ref{['thm:ComplexQuotientnD']}

Key Result

Theorem 1

Let $f,g\in C^\infty(I)$ be smooth real-valued functions on a real interval $I$. Suppose $f,g$ have only simple zeros in $I$ and that their zero sets are identical. Then there exists a non-vanishing smooth function $\varphi\in C^\infty(I)$ such that $f = g\varphi$ and $\varphi=f'/g'$ at the zero lev

Figures (3)

  • Figure 1: The real part of the quotient of $f(x,y)=x+y+\mkern1.5mu\mathbbold{i}\mkern1.5mu y$ and $g(x,y)=x+\mkern1.5mu\mathbbold{i}\mkern1.5mu y$. Blue lines are level lines, showing that $f/g$ takes a constant value along any line passing through the origin.
  • Figure 2: The real part of the quotient of $f(x,y)= x+x^2+\mkern1.5mu\mathbbold{i}\mkern1.5mu x+\mkern1.5mu\mathbbold{i}\mkern1.5mu y$ and $g(x,y) = x + \mkern1.5mu\mathbbold{i}\mkern1.5mu x+\mkern1.5mu\mathbbold{i}\mkern1.5mu y$. The function $f/g$ is continuous but not differentiable at the origin.
  • Figure :

Theorems & Definitions (24)

  • Theorem 1: Smooth quotient of real functions
  • Theorem 2: Smooth quotient of real functions in arbitrary dimension
  • Definition 1: Simple zero
  • Example 1
  • Example 2
  • Example 3
  • Definition 2: Ratio of derivatives
  • Theorem 3: Path dependent l'Hôpital rule
  • Definition 3: Complex linearity
  • Theorem 4: Continuous quotient of complex functions in arbitrary dimension
  • ...and 14 more