TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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A real-differentiable complex map is orientation-preserving conformal at a point exactly when it is complex differentiable there with nonzero derivative

Statement

Let $f : U \to C$ be real totally differentiable at $a \in U$ . Then $f$ is orientation-preserving conformal at $a$ if and only if it is complex differentiable at $a$ and $f^{'} (a) \neq 0$ .

Facts & Assumptions

Given: An open $U \subseteq C$ , a point $a \in U$ , and a map $f : U \to C$ real totally differentiable at $a$ .

[L1]

Complex differentiability at $a$ is equivalent to $D f (a)$ being multiplication by the complex derivative (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[F1]

Orientation-preserving conformality at $a$ means that $D f (a)$ is an orientation-preserving similarity (Orientation-preserving conformality for a real-differentiable complex map at a point).

[L2]

The orientation-preserving similarities of the plane are exactly the maps $h \mapsto ξ h$ with $ξ \neq 0$ (Plane similarities are complex or conjugate-complex multiplications; the orientation-preserving ones are exactly the nonzero complex multiplications).

Proof

technique · direct

1.1

If $f$ is complex differentiable at $a$ with $f^{'} (a) \neq 0$ , [L1] makes $D f (a)$ multiplication by $f^{'} (a)$ , and [L2] makes this an orientation-preserving similarity. Hence $f$ is conformal at $a$ by [F1].

givenL1L2F1

1.2

Conversely, if $f$ is orientation-preserving conformal at $a$ , [F1] and [L2] give $D f (a) h = ξ h$ for some $ξ \neq 0$ . The reverse direction of [L1] makes $f$ complex differentiable with $f^{'} (a) = ξ \neq 0$ .

givenF1L2L1

2.1

Steps 1.1 and 1.2 prove both directions of the equivalence.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 38 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, Corollary 2.2.10 (standard reference, not scraped)
R. Howell and J. Mathews, Complex Analysis, Theorem 9.1.2 (standard reference, not scraped)