A holomorphic function \(f\) is locally injective near \(z_0\) if and only if \(f'(z_0)\ne0\), and then it is conformal at \(z_0\); an injective holomorphic function on a domain has holomorphic inverse on its image. The Möbius transformations \(z\mapsto\frac{az+b}{cz+d}\) with \(ad-bc\ne0\) act simply transitively on ordered triples of distinct points of \(\CC\cup\{\infty\}\) and map circles and lines to circles and lines, and the Cayley transform \(z\mapsto\frac{z-i}{z+i}\) maps the upper half-plane onto \(\DD\). Schwarz lemma: if \(f\colon\DD\to\DD\) is holomorphic with \(f(0)=0\), then \(\abs{f(z)}\le\abs{z}\) for all \(z\in\DD\) and \(\abs{f'(0)}\le1\), and if \(\abs{f(z)}=\abs{z}\) for some \(z\ne0\) or \(\abs{f'(0)}=1\), then \(f(z)=e^{i\theta}z\) for some \(\theta\in\RR\). For \(a\in\DD\), \(\varphi_a(z)=\frac{a-z}{1-\bar az}\) is an automorphism of \(\DD\) with \(\varphi_a\circ\varphi_a=\id\) that exchanges \(a\) and \(0\), and \(\Aut(\DD)=\{e^{i\theta}\varphi_a:\theta\in\RR,\ a\in\DD\}\).
Conformal maps and biholomorphisms (Tag D-PCDNH)
Let \(\Omega\subseteq\CC\) be open. A map \(f\colon\Omega\to\CC\) is conformal on \(\Omega\) if \(f\) is holomorphic on \(\Omega\) and \(f'(z)\neq0\) for every \(z\in\Omega\).
Let \(U,V\subseteq\CC\) be open. A bijective conformal map \(f\colon U\to V\) is a biholomorphism, and \(U\) and \(V\) are biholomorphic, or conformally equivalent, if a biholomorphism \(U\to V\) exists. The biholomorphisms \(\Omega\to\Omega\) form a group under composition, the automorphism group \(\Aut_\CC(\Omega)\).
Let \(\Omega\subseteq\CC\) be open and let \(f\) be conformal on \(\Omega\).
\(f\) preserves signed angles: if \(\gamma_1,\gamma_2\colon(-1,1)\to\Omega\) are \(C^1\) curves with \(\gamma_1(0)=\gamma_2(0)=z_0\) and nonzero tangents \(\gamma_1'(0),\gamma_2'(0)\), then the signed angle from \((f\circ\gamma_1)'(0)\) to \((f\circ\gamma_2)'(0)\) equals the signed angle from \(\gamma_1'(0)\) to \(\gamma_2'(0)\).
If \(U,V\subseteq\CC\) are open and \(f\colon U\to V\) is a holomorphic bijection, then \(f'\) has no zeros and \(f^{-1}\colon V\to U\) is holomorphic and conformal. In particular, the inverse of a biholomorphism is a biholomorphism.
By the chain rule \((f\circ\gamma_j)'(0)=f'(z_0)\,\gamma_j'(0)\). Multiplication by \(f'(z_0)=\abs{f'(z_0)}e^{i\alpha}\neq0\) scales by \(\abs{f'(z_0)}>0\) and rotates by \(\alpha\), adding \(\alpha\) to the argument of both tangent vectors, so the difference of their arguments is unchanged.
A holomorphic bijection \(U\to V\) is univalent, so by the proposition on that card \(f'\) has no zeros and \(f^{-1}\) is holomorphic. Differentiating \(f(f^{-1}(w))=w\) gives \((f^{-1})'(w)=1/f'(f^{-1}(w))\neq0\).
Injective holomorphic maps have nonvanishing derivative (Tag C-FVT4V)
Let \(U,V\subseteq\CC\) be open and let \(f\colon U\to V\) be holomorphic and injective. Then \(f'(z)\neq0\) for every \(z\in U\), and the inverse \(f^{-1}\colon f(U)\to U\) is holomorphic. In particular, the inverse of a conformal map is conformal.
A nowhere-vanishing derivative does not imply injectivity: \(f(z)=e^z\) satisfies \(f'(z)=e^z\neq0\) on \(\CC\), and \(f(0)=f(2\pi i)\).
See [@SS03, Chapter 8, Proposition 1.1].
Möbius transformation (Tag D-FRVBV)
Let \(a,b,c,d\in\CC\) with \(ad-bc\neq0\). The map \[ T(z)\coloneqq\frac{az+b}{cz+d}, \] defined for \(z\in\CC\) with \(cz+d\neq0\), is a linear fractional transformation, or Möbius transformation.
Let \(T(z)=(az+b)/(cz+d)\) with \(ad-bc\neq0\).
\(T\) is a bijection from \(\{z\in\CC:cz+d\neq0\}\) onto \(\{w\in\CC:-cw+a\neq0\}\), with inverse \[ T^{-1}(w)=\frac{dw-b}{-cw+a}. \]
\(T\) is holomorphic on \(\{z\in\CC:cz+d\neq0\}\), with \[ T'(z)=\frac{ad-bc}{(cz+d)^2}\neq0, \] so \(T\) is conformal there.
For \(cz+d\neq0\), \(-c\,T(z)+a=(ad-bc)/(cz+d)\neq0\), and \[ \frac{d\,T(z)-b}{-c\,T(z)+a}=\frac{d(az+b)-b(cz+d)}{-c(az+b)+a(cz+d)}=\frac{(ad-bc)z}{ad-bc}=z. \] The map \(w\mapsto(dw-b)/(-cw+a)\) has coefficients with \(da-bc\neq0\), and the same computation shows that \(T\) inverts it on \(\{w:-cw+a\neq0\}\).
By the quotient rule, \[ T'(z)=\frac{a(cz+d)-c(az+b)}{(cz+d)^2}=\frac{ad-bc}{(cz+d)^2}, \] which is nonzero because \(ad-bc\neq0\).
Three points determine a Möbius transformation (Tag PR-74KHY)
Let \(z_1, z_2, z_3\in\CC\) be distinct. The Möbius transformation \[ T(z) \coloneqq { (z-z_1) (z_2-z_3) \over (z-z_3) (z_2 - z_1)} \] satisfies \(T(z_1)=0\), \(T(z_2)=1\), and \(T(z_3)=\infty\), and it is the unique Möbius transformation with these values. It is denoted \((z; z_1, z_2, z_3)\).
Consequently, for distinct \(w_1,w_2,w_3\in\CC\), the map \[ S\inverseof{\coloneqq(\,\cdot\,; w_1, w_2, w_3)} \circ (\,\cdot\,; z_1,z_2, z_3) \] is the unique Möbius transformation with \(S(z_j)=w_j\) for \(j=1,2,3\).
Substituting \(z_1\), \(z_2\), \(z_3\) into \(T\) gives \(0\), \(1\), \(\infty\). If \(T'\) is another Möbius transformation with the same values, then \(R\coloneqq T'\circ \inverseof{T}\) fixes \(0\), \(1\), and \(\infty\). Fixing \(\infty\) forces \(R(z)=az+b\); fixing \(0\) forces \(b=0\); fixing \(1\) forces \(a=1\); so \(R=\id\) and \(T'=T\). The map \(S\) is a composite of Möbius transformations, and it sends \(z_1,z_2,z_3\) to \(0,1,\infty\) and then to \(w_1,w_2,w_3\). If \(S'\) is another Möbius transformation with \(S'(z_j)=w_j\) for \(j=1,2,3\), then \((\,\cdot\,;w_1,w_2,w_3)\circ S'\inverseof{\circ(\,\cdot\,;z_1,z_2,z_3)}\) fixes \(0\), \(1\), and \(\infty\), so it is the identity by the same argument, and \(S'=S\).
In [@Ahl79, Chapter 3, Section 3.2, Definition 12], the cross ratio \((z_1,z_2,z_3,z_4)\) has four arguments: it is the image of \(z_1\) under the Möbius transformation carrying \(z_2, z_3, z_4\) to \(1, 0, \infty\). In that notation, \((z;z_1,z_2,z_3)=(z,z_2,z_1,z_3)\).
Cayley transform $\frac{i-z}{i+z}$ between the upper half-plane and the disc (Tag T-77SHB)
The fractional linear transformation \[ G(w)=i\,\frac{1-w}{1+w} \] maps the unit disc \(\DD\) biholomorphically onto the upper half-plane \(\HH=\ts{z\st\Im z>0}\). Its inverse is \[ F(z)=\frac{i-z}{i+z}, \] which maps \(\HH\) biholomorphically onto \(\DD\).
For \(w\in\DD\), \(w\neq-1\) and \[ \Im G(w)=\Re\frac{1-w}{1+w}=\frac{\Re\big((1-w)(1+\bar w)\big)}{\abs{1+w}^2}=\frac{1-\abs w^2}{\abs{1+w}^2}>0. \] For \(z\in\HH\), \(z\) is closer to \(i\) than to \(-i\), so \(\abs{F(z)}<1\). Solving \(z=i\frac{1-w}{1+w}\) gives \(z+zw=i-iw\), so \(w(z+i)=i-z\) and \(w=F(z)\); hence \(F\) and \(G\) are mutually inverse holomorphic maps.
Standard conformal maps between half-planes, sectors, strips and discs (Tag T-2KGOX)
Write \(\HH=\ts{\Im z>0}\) for the upper half-plane, \(\DD^+=\ts{\abs z<1,\ \Im z>0}\) for the upper half-disc, \(Q_1=\ts{\Re z>0,\ \Im z>0}\) for the first quadrant, \(S_\beta=\ts{re^{i\theta}\st r>0,\ 0<\theta<\beta}\) for the sector of opening \(\beta\), \(\Sigma=\ts{0<\Im z<\pi}\) for the horizontal strip, and \(\Log\) for the principal branch of the logarithm. Each of the following maps is a biholomorphism from the stated domain onto the stated target.
| Domain | Target | Map |
|---|---|---|
| \(\CC\) | \(\CC\) | translation \(z\mapsto z+h\), \(h\in\CC\) |
| \(\CC\) | \(\CC\) | dilation \(z\mapsto cz\), \(c>0\) |
| \(\CC\) | \(\CC\) | rotation \(z\mapsto e^{i\theta}z\), \(\theta\in\RR\) |
| \(S_{\pi/n}\), \(n\ge1\) | \(\HH\) | \(z\mapsto z^n\) (PR-PDYJC) |
| \(\DD^+\) | \(Q_1\) | \(z\mapsto\frac{1+z}{1-z}\) (PR-PW4Z6) |
| \(\HH\) | \(\Sigma\) | \(z\mapsto\Log z\) (PR-XCDL5) |
| \(\DD^+\) | \(\ts{\Re w<0,\ 0<\Im w<\pi}\) | \(z\mapsto\Log z\) (PR-PELLF) |
| \(\ts{0<\Re z<\pi,\ \Im z>0}\) | \(\DD^+\) | \(z\mapsto e^{iz}\) |
| \(\DD^+\) | \(\HH\) | \(z\mapsto-\frac12\big(z+\frac1z\big)\) (PR-OTMIR) |
| \(\ts{-\frac\pi2<\Re z<\frac\pi2,\ \Im z>0}\) | \(\HH\) | \(z\mapsto\sin z\) (PR-3LBLV) |
Blaschke factors and finite Blaschke products (Tag D-MFPYG)
For \(a\in\DD\), the Blaschke factor, or hyperbolic translation, at \(a\) is \[ \psi_a(z)\coloneqq\frac{a-z}{1-\bar a z},\qquad z\in\CC,\ \bar a z\neq1. \] For \(n\ge1\), points \(a_1,\ldots,a_n\in\DD\), and \(\lambda\in\CC\) with \(\abs{\lambda}=1\), the finite Blaschke product with zeros \(a_1,\ldots,a_n\) is \[ B(z)\coloneqq\lambda\prod_{k=1}^{n}\psi_{a_k}(z)=\lambda\prod_{k=1}^{n}\frac{a_k-z}{1-\bar a_k z}. \]
Some authors take the Blaschke factor to be \(\phi_a(z)\coloneqq\frac{z-a}{1-\bar a z}=-\psi_a(z)\); the two conventions differ by the unimodular constant \(-1\), which is absorbed into \(\lambda\) in a Blaschke product.
Let \(a\in\DD\).
For \(\abs{z}\le1\), \(1-\abs{\psi_a(z)}^2=\frac{(1-\abs{a}^2)(1-\abs{z}^2)}{\abs{1-\bar a z}^2}\); hence \(\psi_a(\DD)\subseteq\DD\) and \(\psi_a(S^1)\subseteq S^1\).
\(\psi_a\circ\psi_a=\operatorname{id}\) on \(\overline{\DD}\), so \(\psi_a\) is a biholomorphism \(\DD\to\DD\) exchanging \(0\) and \(a\).
\(\psi_a\) preserves the hyperbolic metric \(\frac{\abs{dz}}{1-\abs{z}^2}\) on \(\DD\): \(\frac{\abs{\psi_a'(z)}}{1-\abs{\psi_a(z)}^2}=\frac{1}{1-\abs{z}^2}\) for \(z\in\DD\).
Every finite Blaschke product \(B\) is holomorphic on a neighborhood of \(\overline{\DD}\), maps \(\DD\) into \(\DD\) and \(S^1\) into \(S^1\), and its zeros in \(\DD\) are exactly \(a_1,\ldots,a_n\).
Since \(\abs{a}<1\) and \(\abs{z}\le1\), \(\abs{\bar a z}<1\), so \(\psi_a\) is holomorphic on the disc \(\abs{z}<1/\abs{a}\), which contains \(\overline{\DD}\).
Expanding, \(\abs{1-\bar a z}^2-\abs{a-z}^2=1+\abs{a}^2\abs{z}^2-\abs{a}^2-\abs{z}^2=(1-\abs{a}^2)(1-\abs{z}^2)\); divide by \(\abs{1-\bar a z}^2\). The right side is positive for \(\abs{z}<1\) and zero for \(\abs{z}=1\).
For \(\abs{z}\le1\), \[ \psi_a(\psi_a(z))=\frac{a(1-\bar a z)-(a-z)}{(1-\bar a z)-\bar a(a-z)}=\frac{z(1-\abs{a}^2)}{1-\abs{a}^2}=z. \] With (i), \(\psi_a\colon\DD\to\DD\) is holomorphic with holomorphic inverse \(\psi_a\), and \(\psi_a(0)=a\), \(\psi_a(a)=0\).
By the quotient rule, \(\psi_a'(z)=-\frac{1-\abs{a}^2}{(1-\bar a z)^2}\); divide by the expression for \(1-\abs{\psi_a(z)}^2\) in (i).
Each factor is holomorphic near \(\overline{\DD}\), has modulus less than \(1\) on \(\DD\) and equal to \(1\) on \(S^1\) by (i), and vanishes on \(\overline{\DD}\) only at its own \(a_k\); multiply, using \(\abs{\lambda}=1\).
Properties of the Blaschke factors $\psi_a$ (Tag PR-ULJAJ)
For \(a\in\DD\) let \(\psi_a(z)=\frac{a-z}{1-\bar az}\) be the Blaschke factor. Then:
\(\psi_a\in\Aut(\DD)\), and \(\psi_a\) maps the unit circle \(S^1\) onto itself;
\(\psi_a(0)=a\) and \(\psi_a(a)=0\);
\(\psi_a\circ\psi_a=\id_\DD\), so \(\psi_a^{-1}=\psi_a\);
\(\psi_a'(z)=\frac{\abs{a}^2-1}{(1-\bar az)^2}\);
\(\psi_a(\lambda z)=\lambda\,\psi_{\bar\lambda a}(z)\) for every \(\lambda\in\CC\) with \(\abs\lambda=1\).
The denominator \(1-\bar az\) is nonzero for \(\abs{z}\le1\), since \(\abs{\bar az}<1\). For \(\abs{z}=1\), \(\abs{1-\bar az}=\abs{\bar z-\bar a}=\abs{a-z}\), so \(\abs{\psi_a(z)}=1\). By the maximum modulus principle \(\abs{\psi_a}<1\) on \(\DD\) (\(\psi_a\) is nonconstant because \(\abs a<1\)). The values at \(0\) and \(a\) are immediate. A direct computation gives \[ \psi_a(\psi_a(z))=\frac{a(1-\bar az)-(a-z)}{(1-\bar az)-\bar a(a-z)}=\frac{z(1-\abs a^2)}{1-\abs a^2}=z, \] so \(\psi_a\) is a bijection of \(\DD\) equal to its own inverse, hence in \(\Aut(\DD)\), and likewise a bijection of \(S^1\). By the quotient rule, \[ \psi_a'(z)=\frac{-(1-\bar az)+\bar a(a-z)}{(1-\bar az)^2}=\frac{\abs a^2-1}{(1-\bar az)^2}. \] Finally, for \(\abs\lambda=1\), \(\lambda\bar\lambda=1\) gives \[ \lambda\,\psi_{\bar\lambda a}(z)=\lambda\,\frac{\bar\lambda a-z}{1-\lambda\bar az}=\frac{a-\lambda z}{1-\bar a\lambda z}=\psi_a(\lambda z). \]
Schwarz lemma (Tag T-DAETF)
Let \(f\colon\DD\to\DD\) be holomorphic with \(f(0)=0\). Then
\(\abs{f(z)}\leq\abs{z}\) for all \(z\in\DD\), and
\(\abs{f'(0)}\leq1\).
If \(\abs{f(z_0)}=\abs{z_0}\) for some \(z_0\in\DD\setminus\{0\}\), or if \(\abs{f'(0)}=1\), then \(f\) is a rotation: there is \(\lambda\in\CC\) with \(\abs{\lambda}=1\) such that \(f(z)=\lambda z\) for all \(z\in\DD\).
Automorphisms of the unit disc as rotated Blaschke factors (Tag T-W26VL)
For \(a\in\DD\) let \(\psi_a(z)\coloneqq\dfrac{a-z}{1-\overline{a}z}\) be the Blaschke factor at \(a\). The group \(\Aut_\CC(\DD)\) of biholomorphisms \(\DD\to\DD\) is \[ \Aut_\CC(\DD)=\{\lambda\psi_a : a\in\DD,\ \lambda\in S^1\}, \] so every automorphism of \(\DD\) is a Blaschke factor followed by a rotation [@SS03].