Introduction
Complex analysis extends calculus to functions of a complex variable, revealing that requiring a function to be differentiable in the complex sense (a much stronger condition than in the real case) forces remarkably powerful and elegant properties — many results here have no real-variable analogue at all.
1. Complex numbers and geometric representation
A complex number z = x + iy (where i² = −1) can be represented as a point (or vector) in the complex plane, with x as the real part and y as the imaginary part. Addition of complex numbers corresponds to vector addition in this plane; the complex conjugate (z̄ = x − iy) and modulus (|z| = √(x²+y²), the distance from the origin) are essential tools, particularly since z·z̄ = |z|² gives a convenient way to compute reciprocals and simplify fractions involving complex numbers.
2. Polar/exponential form, powers, and roots
Any complex number can be written in polar form, z = r(cosθ + isinθ), or equivalently, via Euler’s formula (e^(iθ) = cosθ + isinθ), in exponential form, z = re^(iθ). This form makes multiplication, powers, and roots dramatically simpler than in rectangular (x + iy) form: De Moivre’s theorem, [r(cosθ + isinθ)]ⁿ = rⁿ(cos nθ + i sin nθ), gives powers directly, and the same idea (allowing θ to range over multiples of 2π before dividing by n) yields all n distinct nth roots of a complex number, evenly spaced around a circle in the complex plane.
3. Functions of a complex variable, limits, and continuity
A complex function w = f(z) maps points in one complex plane to points in another; limits and continuity are defined analogously to the real case, but because z can approach a point from infinitely many directions in the plane (not just from the left or right, as on the real line), the limit must agree along every possible path of approach — a substantially stronger requirement than in single-variable real calculus.
4. Analytic functions and the Cauchy–Riemann equations
A function is analytic (or holomorphic) at a point if it is complex-differentiable throughout some neighbourhood of that point. Writing f(z) = u(x,y) + iv(x,y) in terms of real functions u and v, analyticity requires the Cauchy–Riemann equations, ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x, to hold — these equations link the real and imaginary parts so tightly that each one individually must satisfy Laplace’s equation (they are harmonic functions), directly connecting complex analysis back to the Laplace’s equation met in both Physics (Paper II) and Applied Mathematics (Chapter V).
5. Complex integration and Cauchy’s theorems
Complex integration is performed along a path (contour) in the complex plane. Cauchy’s integral theorem states that the integral of an analytic function around any closed contour is zero, provided the function is analytic everywhere inside and on that contour — a remarkably strong result with no simple real-variable analogue. The related Cauchy integral formula expresses the value of an analytic function at any interior point of a contour purely in terms of an integral of the function’s values on the contour’s boundary — meaning an analytic function’s values everywhere inside a region are already completely determined by its values on the region’s boundary alone.
6. Series, singularities, and residues
Analytic functions can be represented locally by a Taylor series (where the function is analytic) or, near an isolated singularity (a point where the function fails to be analytic), by a more general Laurent series, which allows negative-power terms. The coefficient of the specific 1/(z − z₀) term in this Laurent series is called the residue at that singularity, and the residue theorem states that a contour integral of a function equals 2πi times the sum of the residues at all singularities enclosed by the contour — turning a potentially difficult integral into a matter of identifying singularities and computing residues, a technique frequently used (beyond pure mathematics itself) to evaluate certain real definite integrals that resist elementary real-variable methods.
Key Takeaways
- The complex plane represents complex numbers geometrically; polar/exponential form (via Euler’s formula) dramatically simplifies multiplication, powers (De Moivre’s theorem), and root-finding compared to rectangular form.
- Complex limits must agree along every possible path of approach in the plane, a stronger condition than real-variable limits along a line.
- The Cauchy–Riemann equations are the defining condition for analyticity, and force both the real and imaginary parts of an analytic function to satisfy Laplace’s equation.
- Cauchy’s integral theorem (contour integral of an analytic function is zero) and the Cauchy integral formula (boundary values determine all interior values) are foundational, real-variable-calculus-defying results.
- The residue theorem converts contour integration into identifying singularities and summing their residues, a powerful technique also used to evaluate otherwise difficult real integrals.