[Conformal maps](/page/Conformal%20Maps) are the functions that let complex analysis behave like geometry. A holomorphic function can stretch, rotate, fold, or collapse small figures; the conformal condition isolates the case where, at each point, the first-order behaviour is only a rotation together with a positive change of scale. This is why conformal maps are central in complex analysis, [Holomorphic Function](/page/Holomorphic%20Function), [Derivative](/page/Derivative), and harmonic function theory: they transfer local analytic information while preserving oriented angles. The word "oriented" matters: anti-holomorphic maps preserve the size of angles but reverse their orientation, so they belong to a neighbouring convention rather than to the holomorphic one used here.

The need for the nonvanishing derivative condition is already visible from the map $z \mapsto z^2$. Away from $0$ its first-order behaviour is multiplication by $2z$, so small angles are preserved at nonzero points. At $0$, the derivative vanishes and two distinct tangent directions can collapse into the same first-order direction. Conformal maps remove precisely this kind of critical behaviour.

## Definition

Most applications need a map that has no critical points anywhere on its domain. Requiring a nonzero complex derivative at every point makes the map usable as a system of local complex coordinates throughout the open set, rather than merely as a differentiable function with good behaviour at isolated points. The domain is taken to be open because [complex differentiability](/page/Complex%20Differentiability) is defined through limits from nearby points in every direction.

[definition: Conformal Map] Let $U \subset \mathbb C$ be an open set. A function $f: U \to \mathbb C$ is a conformal map on $U$ if $f$ is holomorphic on $U$ and \begin{align*} f'(w) \ne 0 \end{align*} for every $w \in U$. [/definition]

The global definition is built from a local test. Isolating the pointwise condition is useful because failures of conformality usually occur at critical points, and examples such as $z \mapsto z^2$ fail at a single point while remaining conformal elsewhere.

[definition: Conformal at a Point] Let $U \subset \mathbb C$ be an open set, let $w \in U$, and let $f: U \to \mathbb C$ be holomorphic on a neighbourhood of $w$. The map $f$ is conformal at $w$ if \begin{align*} f'(w) \ne 0. \end{align*} [/definition]

A conformal map need not be globally injective. The exponential function on a horizontal strip of width greater than $2\pi$ is locally conformal everywhere but identifies points differing by $2\pi i$. Because global one-to-one behaviour is often the relevant notion in classifying domains, it deserves its own definition.

[definition: Conformal Isomorphism] Let $U,V \subset \mathbb C$ be open sets. A conformal isomorphism from $U$ to $V$ is a bijective conformal map $f: U \to V$. [/definition]

The definition of conformal isomorphism would be too weak if it only described the forward map. In applications, a conformal change of coordinates is meant to be reversible: a function, contour, or boundary value problem should be transported to the target domain and then brought back without leaving holomorphic geometry. Bijectivity alone gives a set-theoretic inverse, but the analytic content is that this inverse is holomorphic and has no critical points.

[quotetheorem:8269]

This closure under inverses is why conformal isomorphisms are the correct equivalences in planar complex geometry. They identify domains that have the same complex-analytic structure even when their Euclidean shapes look different.

Conformality preserves angles but not lengths. To measure the allowed length distortion, one separates the derivative into a rotational part and a positive size. The positive size is the quantity that appears when arc length, area, or the Laplacian is transformed.

[definition: Conformal Scale Factor] Let $U \subset \mathbb C$ be an open set and let $f: U \to \mathbb C$ be conformal on $U$. The conformal scale factor of $f$ is the function $\lambda_f: U \to (0,\infty)$ defined by \begin{align*} \lambda_f(w) = |f'(w)|. \end{align*} [/definition]

The scale factor records metric distortion, while the argument of $f'(w)$ records rotation. Together they recover the first-order action of $f$ near $w$ as multiplication by the complex number $f'(w)$.

For a compact first model, take the maps whose derivative is the same nonzero complex number everywhere. They show the definition with no hidden global complication.

[example: Nonzero Complex Affine Map] Let $a,b\in\mathbb C$ with $a\ne 0$, and define $f:\mathbb C\to\mathbb C$ by $f(z)=az+b$. For any $z\in\mathbb C$ and any nonzero $h\in\mathbb C$, \begin{align*} \frac{f(z+h)-f(z)}{h}=\frac{a(z+h)+b-(az+b)}{h}=\frac{ah}{h}=a. \end{align*} Hence the limit as $h\to 0$ is $a$, so $f'(z)=a$ for every $z\in\mathbb C$. Since $a\ne 0$, the derivative is nonzero at every point, and therefore $f$ is conformal on $\mathbb C$. Its conformal scale factor is \begin{align*} \lambda_f(z)=|f'(z)|=|a|, \end{align*} so the scale is constant. Multiplication by $a$ rotates tangent directions by $\arg(a)$ and stretches their lengths by $|a|$. To see that $f$ is a conformal isomorphism, define $g:\mathbb C\to\mathbb C$ by $g(w)=(w-b)/a$. Then \begin{align*} g(f(z))=\frac{az+b-b}{a}=z \end{align*} for every $z\in\mathbb C$, and \begin{align*} f(g(w))=a\frac{w-b}{a}+b=w-b+b=w \end{align*} for every $w\in\mathbb C$. Thus $g=f^{-1}$, so $f$ is bijective; because $f$ is conformal on $\mathbb C$, it is a conformal isomorphism from $\mathbb C$ to $\mathbb C$. [/example]

This example is the local model for every conformal map: near each point, a conformal map behaves to first order like a nonzero affine map. Nonlinear conformal maps differ because their scale and rotation can vary from point to point.

## Equivalent Characterisations

The definition is short because complex differentiability is already a strong condition. To see the geometry inside it, write a complex function as a map between real planes. If $f(z) = u(x,y) + i v(x,y)$ with $z = x + iy$, then the real derivative at a point is a [linear map](/page/Linear%20Map) from $\mathbb R^2$ to $\mathbb R^2$.