Cauchy’s Theorem and Its Consequences

A real function can be differentiable once and not twice, smooth but not analytic, bounded and non-constant on the whole line. None of this can happen for holomorphic functions. Integrate one around a closed loop and you get zero (Cauchy's theorem); its values inside a disc are determined by its values on the boundary circle (Cauchy's integral formula); and from that formula everything follows at once: holomorphic functions are infinitely differentiable and equal to their power series, bounded entire functions are constant, the maximum of $|f|$ is on the boundary, and a bounded family of holomorphic functions has a convergent subsequence.

Each of these is the first instance of a theme that runs through the rest of the guide: interior estimates (bounds on derivatives from bounds on the function), Liouville theorems (global solutions are rigid), maximum principles and compactness. In the main line these come back for harmonic functions (6A.2 Harmonic Functions), for the heat equation (6A.4 Maximum Principles, 6A.6 Parabolic Regularity) and finally for the Ricci flow itself.

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

• compute integrals along curves, and explain why $\oint \frac{dz}{z} = 2\pi i$ around the origin;
• state and use Cauchy's theorem and its form for deformed contours;
• derive Cauchy's integral formula and the Cauchy estimates, and deduce that holomorphic functions are analytic;
• prove Liouville's theorem, the fundamental theorem of algebra and the maximum modulus principle;
• state Montel's theorem and explain it as Arzelà–Ascoli plus Cauchy estimates;
• describe two-dimensional ideal flow by a complex potential, and read circulation off a contour integral.

2.1Flow around an obstacle

2.2Integrals along curves

Let $\gamma : [a, b] \to \mathbb{C}$ be a piecewise $C^1$ curve. The contour integral of a continuous $f$ along $\gamma$ is

It doesn't depend on how $\gamma$ is parametrised, only on its route and direction. It satisfies the length estimate

If $f$ has a primitive, a holomorphic $F$ with $F' = f$, then $\int_\gamma f\,dz = F(\gamma(b)) - F(\gamma(a))$ by the fundamental theorem of calculus (2A.11 The Riemann Integral), and the integral around any closed curve is zero. So $\oint z^n\,dz = 0$ for every closed curve when $n \geq 0$, and for curves avoiding $0$ when $n \leq -2$. The exception is $n = -1$: around the circle $z = e^{it}$, $0 \leq t \leq 2\pi$,

This single computation is behind most of the chapter. It says $\frac1z$ has no primitive on $\mathbb{C}\setminus\{0\}$, which is the fact from 5A.1 Holomorphic Functions Are Conformal that there is no continuous logarithm there: the integral measures how much $\log z$ changes once around.

2.3Cauchy's theorem

The same proof applies to a region with holes, as long as its boundary is traversed with the region on the left: outer boundary counterclockwise, inner boundaries clockwise. This gives the deformation principle: if $\gamma_1$ and $\gamma_2$ are two loops around a hole, and $f$ is holomorphic in the region between them, then

(Figure 2.2). You can move a contour freely across any region where the function is holomorphic. That is the circulation statement of the flow example, and it is the main computational tool of 5A.3 Residues and Fourier Transforms.

2.4Cauchy's integral formula

This formula is the heart of the subject. It says that the values of $f$ inside the disc are an average of its values on the boundary, with an explicit kernel. Three consequences follow immediately.

Holomorphic implies analytic. Expand $\frac{1}{\zeta - z} = \frac{1}{\zeta - z_0}\cdot\frac{1}{1 - \frac{z - z_0}{\zeta - z_0}}$ as a geometric series (2A.7 Series), which converges uniformly for $\zeta \in C$ when $|z - z_0| < R$, and integrate term by term:

The power series converges on every disc about $z_0$ that fits inside the domain. Compare the real line, where $e^{-1/x^2}$ is smooth but not analytic and $\frac{1}{1 + x^2}$ is analytic on all of $\mathbb{R}$ but its Taylor series at $0$ has radius $1$ (2B.6 Power Series, Exponentials and Bump Functions). Complex analysis explains the second fact: $\frac{1}{1 + z^2}$ has poles at $\pm i$, at distance $1$ from $0$.

Cauchy estimates. If $|f| \leq M$ on the circle of radius $R$ about $z_0$, then

by the length estimate applied to the derivative formula. A bound on the function on a disc bounds all its derivatives at the centre, with the $n$-th derivative scaling like $R^{-n}$. This is the prototype of an interior estimate.

Mean value property. Taking $z = z_0$ and $\zeta = z_0 + Re^{i\theta}$ in the formula gives

Taking real parts, the same holds for harmonic functions, which 6A.2 Harmonic Functions proves in every dimension.

2.5Liouville, algebra and the maximum modulus

A function holomorphic on all of $\mathbb{C}$ is called entire.

Compare $\sin x$, bounded and non-constant on $\mathbb{R}$: on $\mathbb{C}$ it is unbounded, since $\sin(iy) = i\sinh y$.

The same argument, run for a real harmonic function, is the strong maximum principle of 6A.4 Maximum Principles. It is a cousin of the first maximum principle of 2A.9 Continuous Functions: an interior maximum forces something to be degenerate.

Two further consequences round out the picture. The identity theorem: if two holomorphic functions on a connected open set agree on a set with a limit point (for instance on a small segment), they agree everywhere, because their difference has a power series with all coefficients zero at that limit point, and the set where all derivatives vanish is open and closed. So a holomorphic function is determined by its values on any tiny piece of its domain, which is why the complex extensions of $e^x$, $\sin x$ and $\frac{1}{1 + x^2}$ are unique. Morera's theorem is the converse of Cauchy's: a continuous $f$ whose integral around every triangle is zero is holomorphic, because it has a primitive $F$, which is holomorphic, and $f = F'$ is then holomorphic by Theorem 2.2. Since uniform limits preserve integrals, it follows that a locally uniform limit of holomorphic functions is holomorphic, which fails badly for differentiable real functions (Weierstrass's continuous nowhere-differentiable function is a uniform limit of polynomials, 2B.5 Uniform Convergence and Arzelà–Ascoli).

2.6Montel's theorem: compactness for free

For real functions a uniform bound gives no compactness at all: $\sin(nx)$ is bounded on $[0, 2\pi]$ and has no uniformly convergent subsequence. For holomorphic functions the bound on the function is a bound on the derivative, and compactness comes for free. This is the pattern of every compactness theorem in geometric analysis: an a priori bound, plus an interior estimate upgrading it to derivatives, plus Arzelà–Ascoli. In 5A.4 Harmonic Functions and Conformal Mapping, Montel's theorem produces the Riemann map as the extremal member of a normal family; in Hamilton's compactness theorem (11B.3 Compactness of Ricci Flows) a curvature bound, Shi's estimates and an injectivity radius bound play the same three roles.

2.7History

Cauchy communicated the first general form of his integral theorem to the Paris Academy in 1825, assuming a continuous derivative throughout; the integral formula followed in 1831. Édouard Goursat gave a proof in 1884 by subdividing the region into small squares, and in 1900 removed the assumption that $f'$ is continuous, giving the statement now called the Cauchy–Goursat theorem. The theorem on bounded entire functions was published by Cauchy in 1844; it carries Joseph Liouville's name because he presented it, in connection with doubly periodic functions, in his Paris lectures of 1847. Giacinto Morera's converse dates from 1886. Paul Montel began the study of what he later called normal families in his 1907 thesis, and introduced the name in 1912. The use of complex functions for planar ideal flow goes back to d'Alembert and Euler (5A.1 Holomorphic Functions Are Conformal); the cylinder with circulation is the model from which Kutta and Joukowsky derived lift.

2.8Exercises

Solution

With $z = a + re^{it}$, the integral is $\int_0^{2\pi}r^ne^{int}\cdot ire^{it}\,dt = ir^{n+1}\int_0^{2\pi}e^{i(n+1)t}\,dt$, which is $2\pi i$ for $n = -1$ and $0$ otherwise.

Solution

$(u - iv)(dx + i\,dy) = u\,dx + v\,dy + i(u\,dy - v\,dx)$. The first is the velocity dotted with the tangent; for a counterclockwise curve, $(dy, -dx)$ is the outward normal times arc length, so the second is the flux. For the cylinder, $w = F' = V(1 - a^2/z^2) + \frac{i\Gamma}{2\pi z}$; the first term integrates to $0$ and the second to $\frac{i\Gamma}{2\pi}\cdot2\pi i = -\Gamma$. So the counterclockwise circulation is $-\Gamma$, that is, a clockwise circulation $\Gamma$, and the flux is $0$.

Solution

(a) $2\pi ie$. (b) $\frac{2\pi i}{2!}\cos''(0) = -\pi i$. (c) $\frac{1}{z^2 + 1} = \frac{1}{2i}\big(\frac{1}{z - i} - \frac{1}{z + i}\big)$, so the integral is $\frac{1}{2i}(2\pi i - 2\pi i) = 0$.

Hint

Use the Cauchy estimate for $f^{(k+1)}$ on circles of radius $R \to \infty$.

Solution

$F'(z) = 1 - z^{-2} + \frac{i\Gamma}{2\pi z}$; multiply by $z^2$. On $z = e^{i\theta}$, $F' = 0$ becomes $e^{i\theta} - e^{-i\theta} = -\frac{i\Gamma}{2\pi}$, i.e. $2i\sin\theta = -\frac{i\Gamma}{2\pi}$, which has solutions when $\frac{\Gamma}{4\pi} \leq 1$; the product of the two roots is $-1$, so for $\Gamma < 4\pi$ both have modulus $1$. For $\Gamma > 4\pi$ the roots are $z = -i\big(\frac{\Gamma}{4\pi} \pm \sqrt{\frac{\Gamma^2}{16\pi^2} - 1}\big)$, purely imaginary with product $-1$: one lies outside the cylinder, below it, and the fluid near the cylinder circulates around it on closed streamlines.

Solution

(a) $(u_x)_x = u_{xx} = -u_{yy} = (-u_y)_y$ and $(u_x)_y = u_{xy} = -(-u_y)_x$. An entire function has a primitive (integrate along segments from $0$), and adjusting by a constant, $f = \tilde u + i\tilde v$ with $f' = \tilde u_x - i\tilde u_y = g$; so $\tilde u_x = u_x$ and $\tilde u_y = u_y$, and $\tilde u = u$ after adjusting the real constant. (b) For $\operatorname{Re}f > 0$, $f$ is closer to $1$ than to $-1$, so $|f - 1| < |f + 1|$ and $f + 1 \neq 0$. By Liouville $h$ is constant, so $f = \frac{1 + h}{1 - h}$ is constant, and so is $u$.