Complex Analysis — Basic
The fundamental theory of functions of one complex variable
Undergraduate years 1–2Goals of this level
- Understand the definition of complex differentiation and learn the notion of a holomorphic function.
- Understand the relationship between the Cauchy–Riemann equations and holomorphy.
- Become able to compute complex integrals (integrals along a curve).
- Understand and apply Cauchy's integral theorem and integral formula.
- Learn the theory of power-series expansions and analytic functions.
Prerequisites
- The introductory material (complex numbers, the complex plane, exponential and logarithmic functions)
- Calculus (partial derivatives, the basics of line integrals)
Chapters
Chapter 1: Complex differentiation and holomorphic functions
The definition of complex differentiation, the notion of a holomorphic function, and the strength of differentiability for entire functions.
Chapter 2: The Cauchy–Riemann equations
Deriving the Cauchy–Riemann equations, their equivalence with holomorphy, and harmonic functions.
Chapter 3: Complex integration
The definition of integration along a curve, parametric representation, and basic computational methods.
Chapter 4: Cauchy's integral theorem
Cauchy's integral theorem, simply connected regions, Goursat's theorem, and antiderivatives.
Chapter 5: Cauchy's integral formula
Cauchy's integral formula and its extension to higher derivatives, Morera's theorem, and Liouville's theorem.
Chapter 6: Power series and analytic functions
The radius of convergence of power series, analytic functions, Taylor expansions, and the identity theorem.
Chapter 7: The maximum modulus principle
Cauchy's inequalities, the maximum modulus principle, and the Schwarz lemma.
Chapter 8: Exercises
A problem set for checking your understanding at the basic level.
Supplementary articles
Laurent series
The definition of Laurent series, the annular region of convergence, the classification of singularities, and the relationship with residues.
Entire function
A function holomorphic on the whole complex plane. Polynomials, the exponential, and the trigonometric functions are examples, and it is tied closely to Liouville's theorem.
Euler's formula
The keystone of complex analysis, tying the exponential and trigonometric functions into one. It is the source of why multiplying complex numbers rotates the plane.
Liouville's theorem
The theorem that a bounded entire function must be constant. It also gives a concise proof of the fundamental theorem of algebra.
Morera's theorem
The converse of Cauchy's integral theorem: if the integral along every closed curve is always zero, the function is holomorphic.
Overview
The basic level covers the fundamental theory of complex analysis. The complex analysis treated here centers on the theory of functions of one complex variable, extending the calculus of real numbers to the complex numbers.
What makes complex analysis distinctive is that strikingly beautiful properties hold compared with the real case:
- Once differentiable implies infinitely differentiable: a strong property that does not generally hold for real functions.
- Holomorphic functions can be represented by power series: analytic functions and holomorphic functions coincide.
- Cauchy's integral theorem: an integral along a closed curve is zero.
- Cauchy's integral formula: interior values are determined by boundary values.
At the root of these properties lie the Cauchy–Riemann equations. This system of partial differential equations shows just how strong a constraint complex differentiability is.
What you learn at the basic level becomes the foundation for the intermediate-and-beyond theory, such as the residue theorem and conformal maps.
Reading
The astonishing rigidity of holomorphic functionsReading
Differentiable once means differentiable infinitely often, and the values on a tiny region determine the whole function. A relaxed take on why the "rigidity" of holomorphic functions is beautiful, contrasted with real functions.
Maps that preserve anglesReading
Holomorphic functions deform figures while preserving angles. Why does Greenland look so large on a map? From the story of conformal maps and the Mercator projection, we look at the geometry of complex analysis.
Multiplication makes the world turnReading
Multiplying by a complex number rotates the plane. Why can Euler's formula, which ties the exponential and trigonometric functions into one, describe growth, rotation, and oscillation in the same language? A look at the true nature of "the most beautiful identity."
Frequently asked questions
What does the basic level of complex analysis cover?
Complex numbers and the complex plane, holomorphic functions, the Cauchy–Riemann equations, complex integration and Cauchy's integral theorem, Taylor and Laurent expansions, the residue theorem and its application to real integrals, and the properties of meromorphic functions.
What are the Cauchy–Riemann equations?
They are the necessary and sufficient condition for a complex function $f(z) = u(x,y) + iv(x,y)$ to be holomorphic (complex differentiable): $\partial u/\partial x = \partial v/\partial y$ and $\partial u/\partial y = -\partial v/\partial x$. The real and imaginary parts that satisfy them are both harmonic functions.
What computations is the residue theorem useful for?
Using the "residue" at an isolated singularity of a holomorphic function (the coefficient of the $-1$ power in the Laurent expansion), it computes complex integrals algebraically. It applies powerfully to real integrals such as $\int_{-\infty}^{\infty} \frac{1}{x^4+1} dx$, Fourier transforms, inverse Laplace transforms, and more.