Conformal Mapping, Branches, and the Schwarz Lemma

This page keeps two objects distinct and never silently identifies them.

First, the pointwise principal logarithm of Complex logarithms, the principal logarithm, and principal and multivalued complex powers is defined for every $z\mathrm{\ne }0$ by

$\mathrm{Log}z=\log r+i\theta ,z=r(\cos \theta +i\sin \theta ),-\pi <\theta \le \pi \mathrm{.}$

On the negative real axis the principal polar form uses $\theta =\pi$, so $\mathrm{Log}(-1)=i\pi$. That value is a boundary datum of the slit plane, not the value of a holomorphic branch defined across the cut.

Second, Continuous logarithms and continuous arguments along a contour defines a holomorphic logarithm branch of $z$ on an open set $V$ with $0\mathrm{\notin }V$: a holomorphic $L:V\to \mathbb{C}$ with $\exp (L(z))=z$ for every $z\in V$. On the slit plane $S=\mathbb{C}\setminus \{x\in \mathbb{R}:x\le 0\}$ the principal logarithm is exactly the holomorphic branch normalised by $F(1)=0$, by The principal logarithm is the normalised holomorphic branch on the slit plane; in particular $-\pi <\mathrm{Im}\mathrm{Log}z<\pi$ for every $z\in S$.

Consequently "the principal branch of the logarithm" on this page means the holomorphic function on $S$, while the pointwise principal value on the negative axis is a separate quantity: as $z$ approaches $-1$ from above or below, $\mathrm{Im}\mathrm{Log}z$ tends to $\pi$ or $-\pi$, one-sided limits that assign no value on the cut itself. The branch-defined powers below take a holomorphic branch as input and inherit this discipline.

Let $V\subseteq \mathbb{C}$ be open with $0\mathrm{\notin }V$, and let $L:V\to \mathbb{C}$ be a holomorphic logarithm branch of $z$ on $V$: $L$ is holomorphic and $\exp (L(z))=z$ for every $z\in V$, the branch vocabulary being that of the dictionary in Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers. For $\alpha \in \mathbb{C}$ define the branch power

$z_L^{\alpha }:=\exp (\alpha L(z))(z\in V)\mathrm{.}$

The subscript records the branch: the same base $z$ can carry many holomorphic logarithm branches, and different branches give different values in general. The defining expression is well formed because the complex exponential is a total function on $\mathbb{C}$, whose values and addition law are those of $\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential.

The principal branch power is the special case on the slit plane $S=\mathbb{C}\setminus \{x\in \mathbb{R}:x\le 0\}$:

$z_{\mathrm{Log}}^{}:=\exp (\alpha \mathrm{Log}z)(z\in S),$

where $\mathrm{Log}$ is the holomorphic principal branch of the dictionary remark. On $S$ this agrees with the pointwise principal power of the published principal-logarithm definition; on the negative axis the pointwise principal value is still defined while the holomorphic branch power is not, and the two are not silently identified.

Let $V\subseteq \mathbb{C}$ be open with $0\mathrm{\notin }V$ and let $L:V\to \mathbb{C}$ be a holomorphic logarithm branch of $z$ on $V$. For every integer $n\in \mathbb{Z}$ and every $z\in V$, the branch power of Complex powers defined from a holomorphic logarithm branch equals the complex integer power of Integer powers in the complex field:

$z_L^n:=\exp (nL(z))=z^n\mathrm{.}$

In particular $\exp (nL(z))$ is independent of the choice of branch $L$: the right-hand side $z^n$ mentions no logarithm at all.

For $x>0$ and real $\alpha \in \mathbb{R}$, the principal branch power $x_{\mathrm{Log}}^{}:=\exp (\alpha \mathrm{Log}x)$ of Complex powers defined from a holomorphic logarithm branch equals the published real power $x^{\alpha }$ of Real powers for positive bases, with the zero-base positive-exponent convention:

$x_{\mathrm{Log}}^{}=x^{\alpha }(x>0,\alpha \in \mathbb{R})\mathrm{.}$

Let $V\subseteq \mathbb{C}$ be a connected open set with $0\mathrm{\notin }V$, and let $L_1,L_2:V\to \mathbb{C}$ be two holomorphic logarithm branches of $z$ on $V$. There is a unique integer $k\in \mathbb{Z}$ with

$L_1(z)-L_2(z)=2\pi ik\text{for every }z\in V\mathrm{.}$

Consequently, for every $\alpha \in \mathbb{C}$ the branch powers of Complex powers defined from a holomorphic logarithm branch differ by

$z_{L_1}^{\alpha }=e^{2\pi i\alpha k}{z}_{L_2}^{\alpha }(z\in V)\mathrm{.}$

Additive and multiplicative branch laws are subject to exactly this discrepancy: a claimed identity $\log (zw)=\log z+\log w$ or $(z^{\alpha }{)}^{\beta }=z^{\alpha \beta }$ between branch values holds only after the relevant discrepancy vanishes on the points involved, and the companion page exhibits the principal-branch failures.

Let $U,V\subseteq \mathbb{C}$ be complex domains. $U$ and $V$ are conformally equivalent when there exists a biholomorphism $f:U\to V$ in the sense of Biholomorphic maps between complex domains; such an $f$ is then a conformal equivalence from $U$ onto $V$.

For a complex domain $U$, the automorphism group of $U$ is

$\mathrm{Aut}(U):=\{f:U\to U:f\text{ is biholomorphic}\},$

with composition as the group operation.

Why the group operation is legitimate. The identity map ${\mathrm{id}}_U$ is biholomorphic. If $f$ is biholomorphic then its inverse $f^{-1}$ is holomorphic by the definition of biholomorphy, so $f^{-1}\in \mathrm{Aut}(U)$ when $f\in \mathrm{Aut}(U)$. If $f,g\in \mathrm{Aut}(U)$ then the composite $g\circ f$ is biholomorphic: it is a bijection whose inverse $f^{-1}\circ g^{-1}$ is a composite of holomorphic maps, hence holomorphic. Composition of maps is associative, so these three closure facts make $\mathrm{Aut}(U)$ a group with identity ${\mathrm{id}}_U$.

Fix the unit disc and the upper half-plane

$\mathbb{D}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\},\mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\},$

using modulus and imaginary part of Real and imaginary parts, complex conjugation, and modulus.

For $a\in \mathbb{D}$ define the Blaschke factor

${\phi }_a(z):=\frac{a-z}{1-\bar{a}z}\mathrm{.}$

For every $z$ with $\mathrm{\mid }z\mathrm{\mid }\le 1$, one has $\mathrm{\mid }\bar{a}z\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }<1$, so the denominator never vanishes on the closed unit disc. Thus ${\phi }_a$ is holomorphic on an open disc containing the closed unit disc and in particular on $\mathbb{D}$. Direct substitution records

${\phi }_a(0)=a,{\phi }_a(a)=0.$

For each $a\in \mathbb{D}$, the Blaschke factor

${\phi }_a(z)=\frac{a-z}{1-\bar{a}z}$

is a biholomorphic self-map of $\mathbb{D}$. More precisely,

${\phi }_a(\mathbb{D})=\mathbb{D}\text{and}{\phi }_a({\phi }_a(z))=z(z\in \mathbb{D}),$

so ${\phi }_a$ is an automorphism of the disc.

Let $f:\mathbb{D}\to \mathbb{D}$ be holomorphic and satisfy $f(0)=0$. Then

$\mathrm{\mid }f(z)\mathrm{\mid }\le \mathrm{\mid }z\mathrm{\mid }(z\in \mathbb{D}),\mathrm{\mid }{f}^{\mathrm{\prime }}(0)\mathrm{\mid }\le 1.$

Moreover, if either $\mathrm{\mid }f(z_0)\mathrm{\mid }=\mathrm{\mid }{z}_0\mathrm{\mid }$ for some $z_0\in \mathbb{D}\setminus \{0\}$ or $\mathrm{\mid }{f}^{\mathrm{\prime }}(0)\mathrm{\mid }=1$, then

$f(z)=e^{i\theta }z$

for some real $\theta$; conversely every rotation $z\mapsto e^{i\theta }z$ satisfies equality in both conclusions.

Let $f:\mathbb{D}\to \mathbb{D}$ be holomorphic. Then for every $a,z\in \mathbb{D}$,

$\mid \frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}\mid \le \mid \frac{z-a}{1-\bar{a}z}\mid \mathrm{.}$

Equivalently,

$\mathrm{\mid }{\phi }_{f(a)}(f(z))\mathrm{\mid }\le \mathrm{\mid }{\phi }_a(z)\mathrm{\mid }\mathrm{.}$

Moreover,

$\mathrm{\mid }{f}^{\mathrm{\prime }}(a)\mathrm{\mid }\le \frac{1-\mathrm{\mid }f(a){\mathrm{\mid }}^2}{1-\mathrm{\mid }a{\mathrm{\mid }}^2}(a\in \mathbb{D}),$

and if equality holds for some distinct $a,z\in \mathbb{D}$ or in the derivative inequality at some $a\in \mathbb{D}$, then $f$ is an automorphism of $\mathbb{D}$.

A holomorphic map $f:\mathbb{D}\to \mathbb{D}$ is an automorphism of $\mathbb{D}$ if and only if there exist $a\in \mathbb{D}$ and $\theta \in \mathbb{R}$ such that

$f(z)=e^{i\theta }{\phi }_a(z)=e^{i\theta }\frac{a-z}{1-\bar{a}z}(z\in \mathbb{D})\mathrm{.}$

A map $f:\mathbb{H}\to \mathbb{H}$ is an automorphism of the upper half-plane if and only if

$f(z)=\frac{az+b}{cz+d}$

for real numbers $a,b,c,d$ with $ad-bc>0$.

On the unit disc $\mathbb{D}$ of The unit disc, the upper half-plane, and Blaschke factors fix the Poincare metric, also called the hyperbolic metric of the disc,

$d{s}_{\mathbb{D}}:=\frac{2\mathrm{\mid }dz\mathrm{\mid }}{1-\mathrm{\mid }z{\mathrm{\mid }}^2}\mathrm{.}$

For a piecewise $C^1$ curve $\gamma :[a,b]\to \mathbb{D}$ its Poincare length is

${\mathrm{\ell }}_{\mathbb{D}}(\gamma ):={\int }_{\gamma }\frac{2\mathrm{\mid }dz\mathrm{\mid }}{1-\mathrm{\mid }z{\mathrm{\mid }}^2}={\int }_a^b\frac{2\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(t)\mathrm{\mid }}{1-\mathrm{\mid }\gamma (t){\mathrm{\mid }}^2}dt,$

and the Poincare distance between $z,w\in \mathbb{D}$ is

$d_{\mathbb{D}}(z,w):=\inf \{{\mathrm{\ell }}_{\mathbb{D}}(\gamma ):\gamma \text{ a piecewise }{C}^1\text{ curve in }\mathbb{D}\text{ from }z\text{ to }w\}\mathrm{.}$

The image of $\gamma$ is a compact subset of $\mathbb{D}$, so $1-\mathrm{\mid }\gamma (t){\mathrm{\mid }}^2$ is bounded away from $0$. On each of the finitely many $C^1$ pieces, $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(t)\mathrm{\mid }$ is continuous and bounded; hence the displayed integrand is piecewise continuous and integrable, and each ${\mathrm{\ell }}_{\mathbb{D}}(\gamma )$ is a finite real number. The infimum is taken over a nonempty set, because $\mathbb{D}$ is convex and the segment from $z$ to $w$ lies in $\mathbb{D}$. This normalisation — factor $2$, curvature $-1$ — is fixed for every later surface page. The explicit formula $d_{\mathbb{D}}(z,w)=2\mathrm{artanh}\mathrm{\mid }{\phi }_z(w)\mathrm{\mid }$ and the metric axioms are established on this page by the Poincare-distance formula theorem; the present definition records the intrinsic path-metric construction it evaluates.

For $z,w\in \mathbb{D}$, the Poincare distance on the unit disc satisfies

$d_{\mathbb{D}}(z,w)=2\mathrm{artanh}\mathrm{\mid }{\phi }_z(w)\mathrm{\mid },$

where ${\phi }_z$ is the Blaschke factor carrying $z$ to $0$. Moreover every disc automorphism preserves this distance.

A biholomorphism is holomorphic and has a holomorphic inverse by Biholomorphic maps between complex domains, so it is conformal in the orientation-preserving sense this page uses: holomorphic with nowhere-vanishing derivative. The derivative cannot vanish at any point of its domain, because the local inverse supplied by Holomorphic inverse function theorem and local-degree criterion has derivative $g^{\mathrm{\prime }}(w)=1\mathrm{/}{f}^{\mathrm{\prime }}(g(w))$; a vanishing $f^{\mathrm{\prime }}$ would make that expression undefined. Such a map preserves the magnitude and the orientation of angles between tangent directions at every point.

The convention here is deliberately orientation-sensitive: complex conjugation $z\mapsto \bar{z}$ preserves angle magnitudes but reverses orientation, so it is not conformal in this library's sense. That exclusion is exercised by the companion page's conjugation counterexample.

Conformal equivalence of complex domains is an equivalence relation: every complex domain $U$ is conformally equivalent to itself; if $U$ is conformally equivalent to $V$ then $V$ is conformally equivalent to $U$; and if $U$ is conformally equivalent to $V$ and $V$ to $W$, then $U$ is conformally equivalent to $W$.

Let $n\ge 1$ be an integer, let $\alpha <\beta$ be real numbers with $\beta -\alpha <2\pi \mathrm{/}n$, and define

$S:=\{r{e}^{i\theta }:r>0,\alpha <\theta <\beta \},T:=\{\rho e^{i\varphi }:\rho >0,n\alpha <\varphi <n\beta \}\mathrm{.}$

Then the power map $f(z)=z^n$ is a biholomorphism from $S$ onto $T$.

Let $n\ge 1$ be an integer, let

$S:=\mathbb{C}\setminus \{x\in \mathbb{R}:x\le 0\},V_n:=\{r{e}^{i\theta }:r>0,-\pi \mathrm{/}n<\theta <\pi \mathrm{/}n\},$

and define

$R_n(z):=\exp \text{\hspace{0.17em}⁣}\left(\frac{\mathrm{Log}z}{n}\right)(z\in S)\mathrm{.}$

Then $R_n$ is a biholomorphism from $S$ onto $V_n$, and its inverse is the power map $w\mapsto w^n$ on $V_n$.