Holomorphic Functions Are Conformal

This book is an optional side track. Nothing in the main line, from Book 6A to Perelman, depends on it: harmonic functions, the maximum principle and the two-dimensional Ricci flow all have their own canonical homes (6A.2 Harmonic Functions, 6A.4 Maximum Principles, 11A.7 Ricci Flow on Surfaces). It is here because complex analysis offers a second, very beautiful way of seeing those things, and because it ends at the uniformization theorem, the two-dimensional ancestor of the geometrization conjecture that Ricci flow proved in three dimensions.

The starting point is a single observation. A function $f$ of a complex variable is complex differentiable when its derivative, viewed as a $2 \times 2$ real matrix, is a rotation followed by a scaling. So wherever $f'(z) \neq 0$, a holomorphic map preserves angles: it is conformal. Holomorphic functions are, geometrically, the angle-preserving maps of the plane, and that is why they solve problems in fluid flow, electrostatics and cartography, and why they lead to curvature.

By the end of this chapter you will be able to:

• define complex differentiability and derive the Cauchy–Riemann equations;
• explain why the derivative of a holomorphic function is a rotation plus a scaling, and why holomorphic maps preserve angles where $f' \neq 0$;
• work with the basic examples: powers, $e^z$, the logarithm, Möbius transformations;
• see the Mercator projection as a conformal map, and compute its area distortion;
• use the Joukowsky map to turn flow past a circle into flow past a wing.

1.1A wing from a circle

1.2Complex differentiability

Write $z = x + iy$ and a function of $z$ as $f = u + iv$ with $u, v$ real. Complex numbers multiply by rotating and scaling: multiplying by $re^{i\theta}$ rotates by $\theta$ and scales by $r$ (2B.6 Power Series, Exponentials and Bump Functions).

The definition looks like the one of 2A.10 Derivatives, but the limit is over all directions in the plane at once, and that is a very strong condition. Polynomials in $z$ are holomorphic, with the usual derivatives; so are power series inside their disc of convergence (2B.6 Power Series, Exponentials and Bump Functions), in particular $e^z$; so are quotients where the denominator doesn't vanish. But $f(z) = \bar z$ is not: the difference quotient $\frac{\bar h}{h}$ is $1$ for real $h$ and $-1$ for imaginary $h$.

The matrix $\begin{pmatrix} a & -b\\ b & a\end{pmatrix}$ equals $r\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}$ with $a + ib = re^{i\theta}$: the derivative of a holomorphic function is a rotation by $\arg f'(z_0)$ followed by a scaling by $|f'(z_0)|$. Its determinant is $a^2 + b^2 = |f'(z_0)|^2 \geq 0$, so holomorphic maps preserve orientation, and the area of a small region is multiplied by $|f'|^2$ (3A.5 Product Measures and Change of Variables).

A second consequence: if $u$ and $v$ are $C^2$, then $\Delta u = u_{xx} + u_{yy} = v_{yx} - v_{xy} = 0$, and likewise $\Delta v = 0$. The real and imaginary parts of a holomorphic function are harmonic. (We will see in 5A.2 Cauchy’s Theorem and Its Consequences that holomorphic functions are automatically smooth.) This is the bridge from complex analysis to the Laplace equation, developed in 5A.4 Harmonic Functions and Conformal Mapping.

1.3Holomorphic maps preserve angles

A map with this property is conformal. Conversely, an orientation-preserving $C^1$ map of a plane region whose derivative is everywhere a non-zero multiple of a rotation is holomorphic, by Theorem 1.2. So in the plane, conformal = holomorphic with non-vanishing derivative. (Orientation-reversing conformal maps are the conjugates of holomorphic ones.)

Where $f'(z_0) = 0$ conformality fails, in a specific way: if $f(z) - f(z_0)$ vanishes to order $k$ at $z_0$, angles at $z_0$ are multiplied by $k$. For $f(z) = z^2$ at $0$, the two coordinate axes, at right angles, go to the positive and negative real axis, at a straight angle (Figure 1.2). This is the mechanism behind the Joukowsky airfoil's cusp.

1.4The basic examples

Powers and exponentials. $z^n$ maps the sector of angle $\frac{2\pi}n$ onto the plane minus a ray, multiplying angles at $0$ by $n$. The exponential $e^z = e^x(\cos y + i\sin y)$ (2B.6 Power Series, Exponentials and Bump Functions) maps horizontal lines to rays from $0$ and vertical lines to circles around $0$, conformally, since $(e^z)' = e^z \neq 0$. It maps every horizontal strip of height $2\pi$ onto the plane minus $0$, which is the planar version of the observation in 2B.9 The Inverse and Implicit Function Theorems that a locally invertible map need not be globally injective.

Logarithms. On a region that doesn't wind around $0$, such as the plane minus the negative real axis, $e^z$ has a holomorphic inverse, a branch of the logarithm: $\log z = \log|z| + i\arg z$ with $\arg z \in (-\pi, \pi)$. It maps rays to horizontal lines and circles to vertical segments. There is no continuous logarithm on the whole punctured plane, because going once around $0$ increases the argument by $2\pi$; this is the first appearance of a topological obstruction, the winding number (7A.4 The Fundamental Group), in analysis.

Möbius transformations. For $ad - bc \neq 0$, $T(z) = \frac{az + b}{cz + d}$ is holomorphic except at $z = -d/c$, with $T'(z) = \frac{ad - bc}{(cz + d)^2} \neq 0$. These maps send circles and lines to circles and lines (Exercise 1.6), and compose like $2 \times 2$ matrices. Two of them are used constantly: the Cayley map $z \mapsto \frac{z - i}{z + i}$, a conformal bijection from the upper half-plane onto the unit disc, and the disc automorphisms $z \mapsto e^{i\theta}\frac{z - a}{1 - \bar az}$ with $|a| < 1$, which map the unit disc conformally onto itself. In 5A.4 Harmonic Functions and Conformal Mapping the Schwarz lemma shows these are all the conformal bijections of the disc.

1.5History

Jean d'Alembert (1752) and Leonhard Euler met the Cauchy–Riemann equations in hydrodynamics, as the conditions for a two-dimensional flow to be incompressible and irrotational. Augustin-Louis Cauchy developed complex integration from 1814, and Bernhard Riemann's 1851 thesis made conformal mapping and the equations named after both men the foundation of the subject. Gauss's 1822 prize essay for the Copenhagen Academy treated conformal maps between surfaces in general, motivated by geodesy. Mercator's map dates from 1569; stereographic projection goes back to antiquity (Hipparchus and Ptolemy). Kutta's and Joukowsky's work on lift dates from 1902 and 1906.

1.6Exercises

Solution

(a) Everywhere, $3z^2$. (b) Only at $0$ (the equations $2x = 0$, $2y = 0$). (c) Nowhere. (d) On $\mathbb{C}\setminus\{0\}$, $-1/z^2$. (e) This is $z^2$: everywhere, $2z$. (f) This is $\bar z$: nowhere.

Solution

For $|z| = 1$, $|1 - \bar az| = |\bar z - \bar a| = |z - a|$. By the maximum modulus principle (5A.2 Cauchy’s Theorem and Its Consequences) or a direct computation, $|\phi_a| < 1$ inside. $\phi_a'(z) = \frac{1 - |a|^2}{(1 - \bar az)^2}$, so $\phi_a'(0) = 1 - |a|^2$ and $\phi_a'(a) = \frac{1}{1 - |a|^2}$.

Solution

$\log w = \log\tan(\frac\pi4 + \frac\phi2) + i\lambda = Y + iX$. Differentiating, $\frac{dY}{d\phi} = \frac{\frac12\sec^2(\frac\pi4 + \frac\phi2)}{\tan(\frac\pi4 + \frac\phi2)} = \frac{1}{2\sin(\frac\pi4 + \frac\phi2)\cos(\frac\pi4 + \frac\phi2)} = \frac{1}{\sin(\frac\pi2 + \phi)} = \sec\phi$. North–south, a length $d\phi$ on the sphere becomes $\sec\phi\,d\phi$; east–west, $\cos\phi\,d\lambda$ becomes $d\lambda$: the same factor $\sec\phi$.

Solution

Near any point, $f'$ is non-vanishing and holomorphic, so (on a disc) $f' = e^g$ with $g$ holomorphic (5A.2 Cauchy’s Theorem and Its Consequences gives the existence of such a $g$ on simply connected regions). Then $|f'| = e^{\operatorname{Re}g}$ and $u = \operatorname{Re}g$, the real part of a holomorphic function, which is harmonic.