Functions of a Complex Variable: Basic Concepts

Functions of a Complex Variable

Motivation: Why extend ℝ to ℂ?

In many problems of real analysis—computing integrals such as ∫₋∞^∞ dx/(1+x²), solving differential equations, potential theory—the answer is surprisingly simple, but obtaining it “directly” is hard. It turns out that all these problems are transparently solved if we extend the real number line to the complex plane. Complex analysis—the theory of functions of a complex variable—reveals the deep structure hidden behind real numbers.

The Complex Plane

A complex number z = x + iy is identified with the point (x, y) of the plane ℝ². Here x = Re z is the real part, y = Im z is the imaginary part, i is the imaginary unit (i² = −1).

Modulus: |z| = √(x² + y²) is the distance from the origin. Argument: arg z = arctg(y/x) is the angle with the positive semi-axis (defined up to 2πk). Conjugate number: z̄ = x − iy. Important: z·z̄ = |z|².

A function f: ℂ → ℂ is a mapping from the plane to itself: f(z) = u(x,y) + iv(x,y), where u and v are real-valued functions of two real variables. For example, f(z) = z² = (x+iy)² = x²−y² + 2ixy, so u = x²−y², v = 2xy.

Differentiability and Cauchy–Riemann Conditions

A function f is differentiable at the point z₀ if the limit f'(z₀) = lim_{h→0} (f(z₀+h)−f(z₀))/h exists as h → 0 along any path in ℂ. This is much stricter than real differentiability: the limit does not depend on the direction.

Taking h → 0 along the real axis (h = Δx) and along the imaginary axis (h = iΔy), and equating the results, we obtain the Cauchy–Riemann (CR) conditions: ∂u/∂x = ∂v/∂y, ∂u/∂y = −∂v/∂x.

If the partial derivatives are continuous: f is differentiable ⟺ CR conditions are satisfied.

Derivative: f'(z) = ∂u/∂x + i∂v/∂x = ∂v/∂y − i∂u/∂y.

Holomorphic Functions

A function f is holomorphic (analytic) in a domain D if it is differentiable at every point in D. This is an extremely strict requirement: in real analysis, infinite differentiability ≠ analyticity (there exist C^∞ functions with zero Taylor series). In the complex case, once differentiable ⟹ infinitely differentiable and expansion into a power series!

Harmonic Functions

From the CR conditions and differentiability of u and v, the following results: Δu = ∂²u/∂x² + ∂²u/∂y² = 0 (Laplace equation).

The function u is called harmonic; v is the conjugate harmonic. Thus, the real and imaginary parts of a holomorphic function are harmonic functions. This connects the theory of functions of a complex variable with potential theory and physics: temperature in a stationary thermal field, electric potential—these are harmonic functions.

Elementary Functions

Exponent: eᶻ = eˣ(cos y + i sin y). It is periodic with period 2πi: e^{z+2πi} = eᶻ. Note: |eᶻ| = eˣ > 0, arg eᶻ = y.

Trigonometric: sin z = (eⁱᶻ − e⁻ⁱᶻ)/(2i), cos z = (eⁱᶻ + e⁻ⁱᶻ)/2. They are unbounded on ℂ!

Logarithm: ln z = ln|z| + i arg z—a multivalued function. The principal value Ln z is chosen when arg z ∈ (−π, π].

Power: z^α = e^{α ln z}—multivalued when α ∉ ℤ.

Numerical Example

Task: Check whether f(z) = z² is holomorphic. Check whether g(z) = |z|² = x² + y² is holomorphic.

Step 1. For f(z) = z²: u = x² − y², v = 2xy. ∂u/∂x = 2x, ∂v/∂y = 2x ✓ ∂u/∂y = −2y, −∂v/∂x = −2y ✓ The CR conditions are satisfied everywhere → f(z) = z² is holomorphic on ℂ. The derivative f'(z) = 2x + i·2y = 2(x+iy) = 2z—matches the usual differentiation rule.

Step 2. For g(z) = |z|² = x² + y²: u = x² + y², v = 0. ∂u/∂x = 2x, ∂v/∂y = 0. CR condition: 2x = 0 → only at x = 0. ∂u/∂y = 2y, −∂v/∂x = 0. CR condition: 2y = 0 → only at y = 0. The CR conditions are satisfied only at the point z = 0. Thus, g(z) = |z|² is not holomorphic (although as a function of (x,y) it is infinitely differentiable!). This illustrates the strictness of the conditions for complex differentiability.

Conclusion: Dependence on z̄ (as in g = zz̄) violates holomorphicity. Holomorphic functions depend only on z, but not on z̄.

Practical Application

Aerodynamics and hydrodynamics: the velocity of an ideal incompressible fluid is described by a pair of harmonic functions (potential and stream function), forming a holomorphic function. Conformal mappings (a consequence of holomorphicity) allow recalculating the flow around complex profiles to simple geometries.

Connection with Other Branches of Mathematics

The theory of functions of a complex variable is closely intertwined with differential equations. Solutions to the Laplace equation and the heat conduction equation in the two-dimensional case are described by harmonic and analytic functions; the classical approach of Jacques Hadamard and Richard Kurth for Dirichlet problems relies on the properties of holomorphic functions and the maximum principle. The method of complex characteristics is used in the analysis of partial differential equations of elliptic type.

With algebra, complex analysis is connected, for example, via Gauss’s theorem on the complete factorization of polynomials over the complex numbers and Rouche’s theorem, which allows counting roots inside a contour. These ideas extend into functional analysis: the spectral theorem for normal operators and the functional calculus (Riesz, von Neumann) are built on analytic functions and contour integrals.

The topological aspect appears in the argument principle and the Riemann–Hurwitz theorem, where the number of zeros and poles is linked to the index of a mapping and the Euler characteristic. This direction is based on the theory of Riemann surfaces, developed by Bernhard Riemann and later Hermann Weyl: multivalued functions (root, logarithm) are interpreted as univalent on appropriate manifolds.

In probability theory, a key bridge is planar Brownian motion and martingales. Through Harnack's theorem and properties of harmonic functions, hitting probabilities are described, and conformal mappings are used in stochastic Loewner evolution (Oded Schramm, 2000s). In numerical analysis, Gaussian quadrature methods, algorithms for computing special functions, and stable schemes for solving singular integral equations (the works of Haakon Lovas, Carlson) are based on representing solutions via complex integrals and expansions.

Historical Note and Development of the Idea

The beginnings of complex analysis go back to the 16th century, when Girolamo Cardano and Rafael Bombelli, while solving cubic equations, encountered “imaginary” numbers. The first meaningful geometric interpretations appeared with Caspar Wessel (1799) and Jean-Robert Argand (1806), although wide recognition was received by Carl Friedrich Gauss’s work “Theoria residuorum biquadraticorum” and his treatment of the complex plane.