Complex analysis · Foundations
A function of a complex variable takes a point of one plane and sends it to a point of another, so it is best thought of as a mapping rather than a graph. Before we can do calculus with such mappings we need a vocabulary for sets in the plane (neighbourhoods, domains, curves) and a careful look at the elementary functions, some of which turn out to have several values at once. This post builds that vocabulary and ends with limits and continuity, which look exactly like their real versions but with \(|\cdot|\) meaning distance in the plane.
Everything rests on distance: \(|z_1 - z_2|\) is the ordinary distance between the points \(z_1\) and \(z_2\) of the plane. "Close to \(z_0\)" means "inside a small disk around \(z_0\)".
The point \(\infty\) is added to the plane so that "\(z\) goes far away in any direction" becomes "\(z\) approaches one point". The resulting extended complex plane \( K \cup \{\infty\} \) can be pictured as a sphere (the Riemann sphere), with \(\infty\) at the north pole: large circles \(|z| = R\) become small circles around that pole.
Let \(E\) be a set of points. With disks, the following notions are defined:
| Notion | Definition | Intuition |
|---|---|---|
| Accumulation point of \(E\) | every neighbourhood of \(z_0\) contains a point \(z_1 \ne z_0\) of \(E\) | points of \(E\) crowd arbitrarily close to \(z_0\) |
| Closed set | contains all its accumulation points | you cannot "escape" \(E\) by a limit |
| Interior point | some disk around \(z_0\) lies in \(E\) | \(z_0\) has room in all directions |
| Boundary point | every disk around \(z_0\) contains points of \(E\) and points not in \(E\) | on the edge |
| Open set | consists only of interior points | no edge points included |
| Bounded set | \( |z| < M \) for all \(z \in E\), some \(M > 0\) | fits in a disk |
The condition \( \dot x^2 + \dot y^2 > 0 \) excludes moments where the moving point stops; at such moments the curve can form a sharp cusp even though \(x(t)\) and \(y(t)\) are smooth.
Intuitively, a simply connected domain has no holes: every loop in it can be shrunk to a point without leaving the domain. An annulus \( 1 < |z| < 2 \) is not simply connected: a loop around the hole encloses points (\(z = 0\)) that are not in the domain. This distinction will be crucial for Cauchy's integral theorem.
In the extended plane, the definition is adjusted: \(G\) is simply connected if for each closed Jordan curve \(\Gamma \subset G\), either the interior or the exterior of \(\Gamma\) lies in \(G\). For example, the exterior \(|z| > 1\) together with \(\infty\) is simply connected on the Riemann sphere (it is a disk around the north pole).
A graph of \(f\) would need four real dimensions (two for \(z\), two for \(w\)), so instead we draw two planes and watch how curves and regions of the \(z\)-plane are transported to the \(w\)-plane. Every complex function is a pair of real functions \(u, v\), but not every such pair is interesting: the special pairs that make \(f\) differentiable (the Cauchy–Riemann equations) are the subject of the next section of the course.
| Function | Definition | Defined for |
|---|---|---|
| Power | \( w = z^n \), \( n \in \mathbb N \) | all \(z\) |
| Polynomial | \( P_n(z) = a_nz^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0 \) | all \(z\) |
| Rational | \( w = \frac{P_n(z)}{Q_m(z)} \) | \(z\) not a zero of \(Q_m\) |
| Exponential | \( e^z = e^x(\cos y + i\sin y) \) | all \(z\) |
| Sine, cosine | \( \sin z = \frac{e^{iz} - e^{-iz}}{2i} \), \( \cos z = \frac{e^{iz} + e^{-iz}}{2} \) | all \(z\) |
| Tangent, cotangent | \( \tan z = \frac{\sin z}{\cos z} \), \( \cot z = \frac{\cos z}{\sin z} \) | \( z \ne (k + \frac12)\pi \); \( z \ne k\pi \) (\(k \in \mathbb Z\)) |
| Hyperbolic | \( \sinh z = \frac{e^z - e^{-z}}{2} \), \( \cosh z = \frac{e^z + e^{-z}}{2} \), \( \tanh z \), \( \coth z \) | \(\cosh z \ne 0\), resp. \(\sinh z \ne 0\) |
The definitions are chosen so that for real \(z\) everything agrees with the familiar real functions, and the familiar identities (\( e^{z_1 + z_2} = e^{z_1}e^{z_2} \), \( \sin^2 z + \cos^2 z = 1 \), addition formulas) still hold. But some properties change dramatically:
Inverse functions of periodic or non-injective functions cannot be single-valued. In the complex plane this happens to the most basic inverses.
The inverse of \( w = z^n \) is \( w = \sqrt[n]{z} \), which has \(n\) values for \(z \ne 0\): with \( z = re^{i\varphi} \),
\[ \sqrt[n]{z} = \sqrt[n]{r}\,e^{i(\varphi + 2k\pi)/n}, \qquad k = 0, 1, \dots, n - 1, \]the vertices of a regular \(n\)-gon.
Why: \( e^w = z \) with \( w = u + iv \) means \( e^u = |z| \) and \( v = \) any angle of \(z\). The modulus fixes \(u = \ln|z|\), but the angle is determined only up to multiples of \(2\pi\), which is the periodicity of \(e^z\) seen from the other side. So the logarithm has infinitely many values, spaced \(2\pi i\) apart.
With the logarithm, every power can be defined through the exponential:
\[ a^z = e^{z\operatorname{Ln}a} \ (a \ne 0), \qquad z^\lambda = e^{\lambda\operatorname{Ln}z}, \qquad \operatorname{Log}_az = \frac{\operatorname{Ln}z}{\operatorname{Ln}a} \ (a \ne 0, 1), \]where in the last formula \(\operatorname{Ln}a\) means one chosen branch. All are generally multivalued.
Solving \( z = \sin w = \frac{e^{iw} - e^{-iw}}{2i} \) for \(w\) is a quadratic equation in \(e^{iw}\): \( e^{2iw} - 2iz\,e^{iw} - 1 = 0 \), so \( e^{iw} = iz \pm\sqrt{1 - z^2} \). Hence
\[ \operatorname{Arcsin}z = \frac1i\operatorname{Ln}\big(iz \pm\sqrt{1 - z^2}\big), \qquad \operatorname{Arctan}z = \frac{1}{2i}\operatorname{Ln}\frac{1 + iz}{1 - iz} \ (z \ne \pm i). \]For each \(z\) this is a polynomial equation of degree \(n\) in \(w\), so (generically) it has \(n\) solutions: an algebraic function is in general \(n\)-valued. Example: \( w^2 - z = 0 \) defines \( w = \sqrt z \).
Follow \( z = re^{i\varphi} \) once around the origin, \(\varphi\) from \(0\) to \(2\pi\):
To work with single-valued functions anyway, one makes a cut (for example along the negative real axis) that curves are not allowed to cross. In the cut plane each choice of value (each branch) is a genuine single-valued, continuous function, e.g. the principal value \(\ln z\).
The definition reads exactly like the real one, but there is a big difference in what it demands. In the real case, \(x\) can approach \(x_0\) only from the left or the right. In the plane, \(z\) can approach \(z_0\) along infinitely many paths (lines, spirals, ...), and the limit must be the same along all of them, just as for functions of two real variables (see the post on limits in several variables).
Also useful: \( f(z) \to A \) if and only if \( u \to \operatorname{Re}A \) and \( v \to \operatorname{Im}A \), since \( \max(|\Delta u|, |\Delta v|) \le |\Delta f| \le |\Delta u| + |\Delta v| \). For example, \( \lim_{z\to\infty}\frac1z = 0 \) and \( \lim_{z\to\infty}\frac{2z + 1}{z - i} = 2 \).
All the elementary functions of Section 4 are continuous wherever they are defined; branches of multivalued functions are continuous in a suitably cut plane (e.g. \(\ln z\) is continuous off the negative real axis, but jumps by \(2\pi i\) across it).
The difference from ordinary continuity: \(\delta\) must work for the whole domain at once, not just near one point. As in the real case, a function continuous on a bounded closed domain is uniformly continuous there (Cantor's theorem).
| Concept | Key point |
|---|---|
| Neighbourhood | contains a disk \(|z - z_0| < \varepsilon\); of \(\infty\): contains \(|z| > R\) |
| Domain | open and connected (polygonal paths) |
| Simply connected | no holes: \(\operatorname{int}\Gamma \subset G\) for closed \(\Gamma \subset G\) |
| Complex function | mapping \(w = u(x, y) + iv(x, y)\) |
| \(e^z\), \(\sin z\), \(\cos z\) | \(e^z\) has period \(2\pi i\); \(\sin\), \(\cos\) unbounded |
| \(\operatorname{Ln}z\) | \( \ln|z| + i\arg z + 2k\pi i \): infinitely many values |
| Branch point | algebraic of order \(n\) (\(\sqrt[n]z\)) or transcendental (\(\operatorname{Ln}z\)) |
| Limit | same along all paths; \(f \to A \iff u, v\) converge |
| Continuity | \(\lim f = f(z_0)\); bounded closed domain \(\Rightarrow\) bounded and uniformly continuous |
Try each one before opening the solution. More solved problems from this topic are in Complex Numbers.
\( \frac1z = \frac{\bar z}{|z|^2} = \frac{x - iy}{x^2 + y^2} \): \( u = \frac{x}{x^2 + y^2} \), \( v = -\frac{y}{x^2 + y^2} \).
\( -8 = 8e^{i\pi} \): \( 2e^{i(\pi + 2k\pi)/3} \), \(k = 0, 1, 2\): \( 1 + i\sqrt3 \), \(-2\), \( 1 - i\sqrt3 \).
\( z = \operatorname{Ln}(-2) = \ln 2 + i\pi + 2k\pi i = \ln 2 + (2k + 1)\pi i \).
\( \sin(x + iy) = \sin x\cosh y + i\cos x\sinh y \); then \( |\sin z|^2 = \sin^2x\cosh^2y + \cos^2x\sinh^2y = \sin^2 x + \sinh^2 y \) (using \(\cosh^2 = 1 + \sinh^2\)). It is zero only when \(\sin x = 0\) and \(y = 0\): \( z = k\pi \), the real zeros only.
Along the real axis: \(1\); along the imaginary axis: \(0\). No.
The annulus is open and connected, so a domain, but not simply connected (the circle \(|z| = 1.5\) encloses \(0\), which is not in it). \( \{|z| > 1\} \cup \{\infty\} \) is simply connected in the extended plane: for every closed curve in it, its exterior (the part containing \(\infty\)) lies in the domain.
Moving from real to complex variables changes the scenery more than the definitions. Limits and continuity are defined with the same \(\varepsilon\)–\(\delta\) words, but approaching a point from infinitely many directions makes limits more demanding. The elementary functions keep their identities but lose some old certainties: \(e^z\) becomes periodic, \(\sin z\) and \(\cos z\) become unbounded, and the inverses (roots, logarithms, \(\operatorname{Arcsin}\)) become multivalued, with branch points where their values get tangled. The topological vocabulary (domains, contours, simple connectivity) is what the rest of complex analysis is built on, starting with derivatives and contour integrals.
Related posts: limits and continuity in several variables, and the Fourier integral (where \(e^{i\lambda x}\) is used constantly).