12 Introduction to Complex Analysis

12.1 2-D Problems in Potential Theory Solved Using Complex Analysis

2-D problems in potential theory can be elegantly solved using complex analysis, the theory of differentiable complex-valued functions of complex variables.

The following tools from mathematics are used:

Cauchy’s Integral Theorem
Cauchy’s Integral Formula
Cauchy-Riemann Differential Equations
Residue Theorem of Complex Analysis

12.2 Space of Complex Numbers \(\mathbb{C}\)

\(\mathbb{C}\) is a two-dimensional real vector space with the canonical basis \((1, i)\). A point \(z \in \mathbb{C}\) has real Cartesian coordinates \(x := \mathrm{Re}(z)\), \(y := \mathrm{Im}(z) \in \mathbb{R}\), briefly expressed as \(z = x + i y\).

A complex-valued function \(f: U \subset \mathbb{C} \to \mathbb{C}\) on an open subset of \(\mathbb{C}\) can therefore be expressed by decomposing it into its real and imaginary parts \(f(x + i y) = u(x, y) + i v(x, y)\), viewed as an \(\mathbb{R}^2\)-valued function of two real variables \((x, y) \in \tilde{U} := \{(a, b) \in \mathbb{R}^2 \ | \ a + i b \in U\}\).

12.3 Cauchy’s Integral Theorem

In real analysis, the value of an integral depends on the integration bounds. For complex functions, the path also matters.

Let \(U \subset \mathbb{C}\) be a simply connected (no holes) open domain (elementary domain), and \(f: U \mapsto \mathbb{C}\) a holomorphic function. Furthermore, let \(\gamma : [a, b] \to U\) be a smooth, closed curve. Then: \[ \oint_{\gamma} f(z) \, \mathrm{d}z = 0. \]

Note

A holomorphic function is a complex-valued function \(f:\mathbb C\to\mathbb C\) that is complex differentiable at every point of an open set in the complex plane.

The key property is complex differentiability in the sense of the limit \[ f’(z_0)=\lim_{h\to 0}\frac{f(z_0+h)-f(z_0)}{h} \] existing for all complex directions of \(h\). This requirement is far stronger than real differentiability and leads to remarkable consequences.

Holomorphic functions preserve angles and local geometric structure in the following precise sense:

Conformal behaviour

Let \[f:\Omega\subset\mathbb C\to\mathbb C\] be holomorphic with a non-vanishing derivative \(f’(z_0)\neq 0\). Then, in a neighbourhood of \(z_0\), the mapping acts like a complex linear map \[ w = f(z) \approx f(z_0) + f’(z_0)(z - z_0). \]

This map consists of two elementary operations:

Rotation: Multiplication by \(e^{i\theta}\) rotates all small line segments by the same angle \(\theta\).
Uniform scaling: Multiplication by \(|a|\) stretches or shrinks all small line segments by the same factor.

There is no shearing, no anisotropic distortion, and no reflection.

Angle preservation

Suppose two smooth curves meet at \(z_0\) under an angle \(\alpha\). Under \(f\), the corresponding tangent vectors are multiplied by the same complex number \(f’(z_0)\), which rotates and scales them uniformly. Therefore, the angle between the curves is unchanged.

Formally, if \(v_1\) and \(v_2\) are tangent directions at \(z_0\), then \[ \arg\!\left(\frac{f’(z_0)v_2}{f’(z_0)v_1}\right) =\arg\!\left(\frac{v_2}{v_1}\right), \] so the orientation of the angle is preserved.

Preservation of local structure

A holomorphic function with \(f’(z_0)\neq 0\) maps a sufficiently small neighbourhood of \(z_0\) onto a small neighbourhood of \(f(z_0)\) in a way that is locally similar: shapes become rotated and uniformly scaled copies of themselves. Circles become circles, small grids become locally rotated/scaled grids, and the mapping remains injective in a small neighbourhood.

This behaviour is what is meant by saying that holomorphic functions are conformal at points where their derivative is non-zero.

Proof: With \(f = u + i v\) and \(\mathrm{d}z = \mathrm{d}x + i \mathrm{d}y\), it follows: \[ \oint_{\gamma} f(z) \, \mathrm{d}z = \oint_{\gamma} (u + i v) (\mathrm{d}x + i \mathrm{d}y) = \oint_{\gamma} (u \mathrm{d}x - v \mathrm{d}y) + i \oint_{\gamma} (v \mathrm{d}x + u \mathrm{d}y). \] Using Green’s Theorem: \[ \iint_D \left(\frac{\partial g}{\partial x}(x,y) - \frac{\partial f}{\partial y}(x,y)\right)\, \mathrm{d}x \, \mathrm{d}y = \oint_{C} \left(f(x,y)\, \mathrm{d}x + g(x,y)\, \mathrm{d}y\right), \] the line integrals are replaced by surface integrals over the region \(D\) enclosed by the curve \(\gamma\): \[ \begin{align} \oint_\gamma (u \, \mathrm{d}x - v \, \mathrm{d}y) & = \iint_D \left( -\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} \right) \, \mathrm{d}x\,\mathrm{d}y, \\ \oint_\gamma (v \, \mathrm{d}x + u \, \mathrm{d}y) & = \iint_D \left( \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \right) \, \mathrm{d}x\,\mathrm{d}y. \end{align} \] For the real and imaginary parts \(u\) and \(v\) of holomorphic functions in the domain \(D\), the Cauchy-Riemann differential equations hold: \[ \begin{align} u_x & = +v_y, \\ u_y & = -v_x. \end{align} \] Thus, both integrands—and consequently the integral—are zero: \[ \iint_D \left( -\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} \right) \, \mathrm{d}x\,\mathrm{d}y = \iint_D \left( \frac{\partial u}{\partial y} -\frac{\partial u}{\partial y} \right) \, \mathrm{d}x\,\mathrm{d}y = 0, \] and \[ \iint_D \left( \frac{\partial u}{\partial x} -\frac{\partial v}{\partial y} \right) \, \mathrm{d}x\,\mathrm{d}y = \iint_D \left( \frac{\partial u}{\partial x} -\frac{\partial u}{\partial x} \right) \, \mathrm{d}x\,\mathrm{d}y = 0. \] Hence: \[ \oint_{\gamma} f(z) \, \mathrm{d}z = 0 \qquad \square \]

12.4 Cauchy’s Integral Formula (CIF)

The Cauchy integral formula is one of the central results of complex analysis. It expresses the value of a holomorphic function inside a closed curve entirely in terms of its boundary values.

Let \(U\) be an open subset of the complex plane \(\mathbb{C}\). Consider the closed disk \(D\) fully contained within \(U\), defined as: \[ D = \{ z : |z - z_{0}| \le r \}. \] Let \(f: U \mapsto \mathbb{C}\) be a holomorphic function, and \(\gamma\) a counterclockwise-oriented circle along the boundary of \(D\). Then, for any point \(a\) inside \(D\): \[ f(a) = \frac{1}{2 \pi i} \oint_{\gamma} \frac{f(z)}{z - a} \, \mathrm{d}z. \] The CIF states that the values of a holomorphic function \(f\) inside a disk are fully determined by its values on the boundary of that disk.

12.5 Residue Theorem

If \(f(z)\) is holomorphic in the neighborhood of a point \(z_{0}\), then: \[ \oint_{C} f(z) \, \mathrm{d}z = 0. \] However, if \(f(z)\) has an isolated singularity at \(z_{0}\), the integral is generally nonzero and satisfies: \[ \frac{1}{2 \pi i} \oint_{C} f(z) \, \mathrm{d}z = \mathrm{Res} f(z). \] For a simple pole: \[ \mathrm{Res} f(z) = \lim_{ z \to z_{0} } (z - z_{0}) f(z). \] Instead of computing contour integrals, one only needs to calculate residues, which is often simpler.