Complex functions, analytic functions, Cauchy-Riemann equations, contour integration, Cauchy's theorem and formula, Laurent series, poles, and the residue theorem — the complete graduate-level guide.
A complex number z is written as z = x + iy, where x = Re(z) is the real part, y = Im(z) is the imaginary part, and i² = −1. The complex plane (Argand plane) represents z as the point (x, y).
| Operation | Formula |
|---|---|
| Addition | (x₁ + iy₁) + (x₂ + iy₂) = (x₁+x₂) + i(y₁+y₂) |
| Subtraction | (x₁ + iy₁) − (x₂ + iy₂) = (x₁−x₂) + i(y₁−y₂) |
| Multiplication | (x₁+iy₁)(x₂+iy₂) = (x₁x₂−y₁y₂) + i(x₁y₂+x₂y₁) |
| Division | (z₁/z₂) = (z₁ · z̄₂) / |z₂|² |
| Powers | zⁿ = rⁿe^(inθ) = rⁿ(cos nθ + i sin nθ) |
| nth Roots | z^(1/n) = r^(1/n) · e^(i(θ+2πk)/n), k = 0,1,…,n−1 |
A complex function f: ℂ → ℂ maps complex numbers to complex numbers. Writing z = x + iy and f(z) = u + iv, we can decompose f into two real-valued functions of two real variables: u(x, y) and v(x, y). The behavior of complex functions is richer and more constrained than real functions.
lim(z→z₀) f(z) = L means: for every ε > 0 there exists δ > 0 such that 0 < |z − z₀| < δ implies |f(z) − L| < ε. The key difference from real limits: z can approach z₀ from any direction in the 2D complex plane, and the limit must be the same regardless of direction.
f is continuous at z₀ if lim(z→z₀) f(z) = f(z₀). Polynomials and rational functions (away from poles) are continuous. The composition of continuous functions is continuous. Key fact: a function continuous on a closed bounded set is uniformly continuous and achieves its maximum modulus on the boundary.
Directional Dependence Warning
f(z) = z̄/z at z = 0: along the real axis, f(x) = x/x = 1; along the imaginary axis, f(iy) = −iy/iy = −1. The limit does not exist because it depends on direction. This behavior is impossible for real limits of one variable and is a key complication in complex analysis.
A function f(z) = u(x, y) + iv(x, y) is complex-differentiable at z₀ = x₀ + iy₀ if the limit [f(z₀ + h) − f(z₀)]/h exists as h → 0 in ℂ. This requires the partial derivatives to satisfy the Cauchy-Riemann (CR) equations.
Theorem: If f = u + iv is differentiable at z₀, then the CR equations hold at (x₀, y₀). Conversely, if u and v have continuous partial derivatives satisfying the CR equations in a neighborhood of z₀, then f is complex-differentiable at z₀.
When CR equations hold, f'(z) can be computed as:
f'(z) = ∂u/∂x + i·∂v/∂x
f'(z) = ∂v/∂y − i·∂u/∂y
For f = u(r,θ) + iv(r,θ):
r·∂u/∂r = ∂v/∂θ
∂u/∂θ = −r·∂v/∂r
f is analytic (holomorphic) at z₀ if it is complex-differentiable in an open neighborhood of z₀. Analyticity is a global property; complex-differentiability at a single point is weaker. An entire function is analytic everywhere in ℂ (e.g., polynomials, e^z, sin z, cos z).
f(z) = z²
u = x²−y², v = 2xy
Analytic everywhere
f(z) = e^z
u = eˣcos y, v = eˣsin y
Entire function
f(z) = z̄
u = x, v = −y
Nowhere analytic
A real-valued function u(x, y) is harmonic on a domain D if it satisfies Laplace's equation throughout D:
If f = u + iv is analytic, then both u and v are harmonic. Proof: differentiate the first CR equation (∂u/∂x = ∂v/∂y) with respect to x, and the second (∂u/∂y = −∂v/∂x) with respect to y, then add. The functions u and v are called harmonic conjugates. Given u harmonic on a simply connected domain, a harmonic conjugate v can always be found (up to a constant) via line integrals of the CR equations.
Physical Applications
Key Properties
A contour (curve) in the complex plane is a piecewise smooth path C: z(t) = x(t) + iy(t) for t ∈ [a, b]. The contour integral of f along C is defined as:
Integrate f(z) = (z − z₀)ⁿ around the circle |z − z₀| = r:
This is the cornerstone of the residue theorem. The n = −1 case is the source of the 2πi factor that appears throughout complex integration.
Circular contour
z = z₀ + re^(iθ), θ ∈ [0, 2π]. For residues at isolated singularities.
Semicircular contour
Upper half-disk with diameter [−R, R]. For real integrals over (−∞, ∞).
Rectangular contour
Axis-aligned rectangle. For integrals of exponentials and periodic functions.
Keyhole contour
Avoids branch cut along positive real axis. For functions like z^α or log z.
Indented contour
Semicircle with small indent around real singularity. For principal value integrals.
Dogbone contour
Two branch cuts with loops. For integrals involving square roots of polynomials.
Conditions
Consequences
For f analytic inside and on C, and z₀ any interior point. This extraordinary result expresses the value of f at any interior point as a weighted average of f on the boundary.
The generalized formula gives all derivatives of f at z₀. Since n can be any positive integer, every analytic function is infinitely differentiable — a striking difference from real analysis, where differentiability does not imply higher differentiability.
Liouville's Theorem
Every bounded entire function is constant. Proof: apply Cauchy's formula to f' with a large circle; the bound forces f' = 0. Corollary: the Fundamental Theorem of Algebra (every non-constant polynomial has a root in ℂ).
Morera's Theorem (Converse)
If f is continuous on a domain D and ∮_C f(z) dz = 0 for every simple closed contour C in D, then f is analytic on D. Useful for proving analyticity without directly checking CR equations.
If f is analytic in a disk |z − z₀| < R, it equals its Taylor series there. The radius of convergence R equals the distance from z₀ to the nearest singularity.
e^z =
1 + z + z²/2! + z³/3! + …
sin z =
z − z³/3! + z⁵/5! − …
cos z =
1 − z²/2! + z⁴/4! − …
1/(1−z) =
1 + z + z² + z³ + … (|z|<1)
Components
Example: e^(1/z) at z = 0
e^(1/z) = 1 + 1/z + 1/(2!z²) + 1/(3!z³) + …
Essential singularity at z = 0
(infinitely many negative powers)
| Type | Principal Part | Behavior as z → z₀ | Example |
|---|---|---|---|
| Removable | None (a₋ₙ = 0 for all n ≥ 1) | f(z) → finite limit | sin(z)/z at z=0 |
| Pole of order m | Finitely many terms, lowest (z−z₀)^(−m) | |f(z)| → ∞ | 1/z² at z=0 (order 2) |
| Simple pole | One term: a₋₁/(z−z₀) | |f(z)| → ∞ like 1/|z−z₀| | 1/(z−1) at z=1 |
| Essential | Infinitely many negative-power terms | Oscillates wildly (Picard) | e^(1/z) at z=0 |
Simple pole at z₀
f(z) = 1/(z²−1) at z=1: Res = lim_{z→1} (z−1)·1/((z−1)(z+1)) = 1/(1+1) = 1/2
Pole of order m at z₀
f(z) = e^z/z² at z=0 (order 2): Res = lim_{z→0} d/dz[z²·e^z/z²] = lim_{z→0} e^z = 1
f(z) = p(z)/q(z), simple zero of q at z₀
f(z) = z/(z²+1) at z=i: Res = i/(2i) = 1/2
Where the sum is over all isolated singularities zₖ inside C. The contour C is simple, closed, and positively oriented (counterclockwise). f must be analytic on C itself. The theorem unifies Cauchy's integral theorem (no singularities → sum = 0) and Cauchy's integral formula (one simple pole contributed by 1/(z−z₀)).
To evaluate ∫ from −∞ to ∞ of p(x)/q(x) dx (degree q ≥ degree p + 2, q has no real roots):
Close contour with semicircle in upper half-plane |z| = R, Im(z) ≥ 0
The semicircular arc → 0 as R → ∞ (Jordan's lemma or ML bound)
By residue theorem: ∫_{−∞}^∞ f(x) dx = 2πi · Σ Res(f, zₖ) for Im(zₖ) > 0
Sum residues at poles in upper half-plane only
Integrals of the form ∫₀^(2π) R(cos θ, sin θ) dθ: substitute z = e^(iθ), dθ = dz/(iz), cos θ = (z + z⁻¹)/2, sin θ = (z − z⁻¹)/(2i). The integral becomes a contour integral on the unit circle |z| = 1.
f(z) = (x+iy)³ = x³ − 3xy² + i(3x²y − y³)
u = x³ − 3xy², v = 3x²y − y³
∂u/∂x = 3x² − 3y² ∂v/∂y = 3x² − 3y² ✓
∂u/∂y = −6xy −∂v/∂x = −6xy ✓
CR equations satisfied everywhere → f(z) = z³ is entire
Derivative: f'(z) = ∂u/∂x + i∂v/∂x = (3x²−3y²) + i(6xy) = 3(x+iy)² = 3z²
Evaluate ∮_C e^z / (z − πi) dz where C is |z| = 4
z₀ = πi is inside C (|πi| = π ≈ 3.14 < 4)
f(z) = e^z is analytic everywhere
By Cauchy's formula: ∮ e^z/(z−πi) dz = 2πi · f(πi)
f(πi) = e^(πi) = cos π + i sin π = −1
Result: 2πi · (−1) = −2πi
f(z) = 1/[z(z−1)] = 1/(z−1) · 1/z
Partial fractions: 1/[z(z−1)] = −1/z + 1/(z−1)
1/(z−1) = −1/(1−z) = −Σ(n=0 to ∞) zⁿ (for |z| < 1)
So: f(z) = −1/z − 1 − z − z² − z³ − …
Residue at z = 0: a₋₁ = −1
Simple pole at z = 0 and z = 1. Valid for 0 < |z| < 1.
Poles of 1/(z⁴+1): at z⁴ = −1 = e^(iπ), so z = e^(iπ(2k+1)/4)
k=0: z₁ = e^(iπ/4) = (1+i)/√2 (upper half-plane)
k=1: z₂ = e^(i3π/4) = (−1+i)/√2 (upper half-plane)
Res(f, z₁) = 1/(4z₁³) = z₁/(4z₁⁴) = z₁/(4·(−1)) = −z₁/4
Res(f, z₂) = −z₂/4
Sum of residues: −(z₁+z₂)/4 = −((1+i)/√2 + (−1+i)/√2)/4 = −(2i/√2)/4 = −i/(2√2)
∫₋∞^∞ = 2πi · (−i/(2√2)) = 2πi · (−i)/(2√2) = π/√2 = π√2/2
Result: π√2/2 ≈ 2.221
The Cauchy-Riemann equations are the conditions a complex function f(z) = u(x, y) + iv(x, y) must satisfy to be complex-differentiable (analytic) at a point. They state: ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x. If f is differentiable as a function of two real variables AND satisfies these equations at a point, then f is complex-differentiable there. They matter because analyticity is an extremely strong condition — it implies the function is infinitely differentiable and equals its Taylor series everywhere in its domain.
Euler's formula states e^(iθ) = cos θ + i sin θ for any real θ. It connects the exponential function to trigonometry via the imaginary unit. In polar form, any complex number z with modulus r and argument θ is written z = r·e^(iθ). The famous identity e^(iπ) + 1 = 0 follows by setting θ = π. Euler's formula makes multiplication of complex numbers geometric: multiplying by e^(iθ) rotates a point in the complex plane by angle θ.
Cauchy's integral theorem states that if f is analytic on and inside a simple closed contour C, then the contour integral ∮_C f(z) dz = 0. The integral of an analytic function around any closed loop enclosing only points where f is analytic is exactly zero. This is analogous to path-independence for conservative vector fields. A key consequence is that the value of a contour integral depends only on the singularities enclosed, not on the specific path taken (as long as homotopy is preserved).
Cauchy's integral formula states: if f is analytic inside and on a simple closed positively-oriented contour C, and z₀ is any point inside C, then f(z₀) = (1/2πi) ∮_C f(z)/(z − z₀) dz. This remarkable result means the values of an analytic function inside a region are completely determined by its values on the boundary. A generalized form gives all higher derivatives: f⁽ⁿ⁾(z₀) = (n! / 2πi) ∮_C f(z)/(z − z₀)^(n+1) dz.
A Taylor series represents an analytic function as a power series Σ aₙ(z − z₀)ⁿ with non-negative integer powers only. It converges in a disk around z₀ and works when f is analytic there. A Laurent series extends this to include negative powers: Σ aₙ(z − z₀)ⁿ where n runs from −∞ to +∞. Laurent series are used when f has a singularity at z₀. The 'principal part' consists of the negative-power terms. If the principal part has finitely many terms, z₀ is a pole; if infinitely many, it's an essential singularity.
The residue of f at an isolated singularity z₀ is the coefficient a₋₁ of the (z − z₀)⁻¹ term in the Laurent series of f at z₀. For a simple pole: Res(f, z₀) = lim(z→z₀) (z − z₀)f(z). For a pole of order m: Res(f, z₀) = (1/(m−1)!) · lim(z→z₀) d^(m−1)/dz^(m−1) [(z−z₀)^m f(z)]. For a removable singularity, the residue is 0. Residues are essential for computing contour integrals via the residue theorem.
The residue theorem states: if f is analytic inside and on a simple closed positively-oriented contour C except at finitely many isolated singularities z₁, z₂, …, zₙ inside C, then ∮_C f(z) dz = 2πi · Σ Res(f, zₖ). The integral equals 2πi times the sum of residues at all enclosed singularities. The residue theorem is a powerful tool for evaluating real definite integrals (by converting them to complex contour integrals), summing series, and computing inverse Laplace and Fourier transforms.
A real-valued function u(x, y) is harmonic if it satisfies Laplace's equation: ∂²u/∂x² + ∂²u/∂y² = 0. If f(z) = u + iv is analytic, then both u and v are harmonic — this follows directly from the Cauchy-Riemann equations. Moreover, u and v are called harmonic conjugates of each other. Given a harmonic function u, one can always find (locally) a harmonic conjugate v to form an analytic function. Harmonic functions model steady-state heat flow, electrostatics, and ideal fluid flow.
An isolated singularity z₀ of f is classified by the principal part of its Laurent series. A removable singularity has no principal part (all negative-power coefficients are 0); f can be redefined at z₀ to make it analytic — example: sin(z)/z at z = 0. A pole of order m has finitely many negative powers, down to (z − z₀)^(−m); f blows up as z → z₀ — example: 1/z² has a pole of order 2 at 0. An essential singularity has infinitely many negative-power terms; f behaves wildly near z₀ (Picard's theorem: f takes every complex value except possibly one). Example: e^(1/z) at z = 0.
Sequences, series, limits, continuity, and differentiation on the real line — the foundation for complex analysis
Partial derivatives, multiple integrals, line integrals, and Green's theorem — prereqs for contour integration
Complex analysis powers Laplace transforms, Fourier analysis, and solutions to PDEs like the heat and wave equations
Interactive problems on contour integrals, residues, and analytic functions — with step-by-step solutions and private tutoring, free to try.
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