Complex Analysis
What is Complex Analysis?
A number of the form x + iy, where x and y are real numbers, and i is the imaginary unit satisfying i² = -1.
Key formula / rule: Complex Number (Cartesian Form)
Key points
- Understand the algebra and geometry of complex numbers.
- Determine if a given function is analytic using Cauchy-Riemann equations.
- Evaluate complex integrals using Cauchy's Integral Theorem and Formula.
- Expand functions into Taylor and Laurent series.
Common exam trap
Incorrectly applying Cauchy-Riemann equations (e.g., mixing partial derivatives).
Definitions
- Term
Complex Number
- Meaning
A number of the form x + iy, where x and y are real numbers, and i is the imaginary unit satisfying i² = -1.
- Term
Analytic Function
- Meaning
A complex function f(z) that is differentiable at every point in an open set. Also known as a holomorphic function.
- Term
Singularity
- Meaning
A point z₀ where a complex function f(z) fails to be analytic.
- Term
Pole
- Meaning
An isolated singularity z₀ of a function f(z) such that lim(z→z₀) |f(z)| = ∞. The order of the pole is the smallest integer m such that lim(z→z₀) [(z-z₀)ᵐ f(z)] is a finite non-zero number.
- Term
Residue
- Meaning
For an isolated singularity z₀ of a function f(z), the residue is the coefficient of 1/(z-z₀) in the Laurent series expansion of f(z) around z₀.
Learning objectives
Understand the algebra and geometry of complex numbers.
Determine if a given function is analytic using Cauchy-Riemann equations.
Evaluate complex integrals using Cauchy's Integral Theorem and Formula.
Expand functions into Taylor and Laurent series.
Identify different types of singularities and calculate residues.
Apply the Residue Theorem to evaluate complex and certain real integrals.
Formulae
- Name
Complex Number (Cartesian Form)
- Note
x is the real part, y is the imaginary part.
- Expression
z = x + iy
- Name
Complex Number (Polar Form)
- Note
r = |z| = √(x² + y²) (modulus), θ = arg(z) (argument).
- Expression
z = r(cosθ + i sinθ)
- Name
Complex Number (Exponential Form)
- Note
Derived from Euler's formula e^(iθ) = cosθ + i sinθ.
- Expression
z = r e^(iθ)
- Name
Cauchy-Riemann Equations
- Note
Conditions for f(z) = u(x,y) + iv(x,y) to be analytic.
- Expression
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
- Name
Cauchy's Integral Theorem
- Note
If f(z) is analytic inside and on a simple closed contour C.
- Expression
∮C f(z) dz = 0
- Name
Cauchy's Integral Formula
- Note
For f(z) analytic inside and on C, and z₀ inside C.
- Expression
f(z₀) = 1/(2πi) ∮C f(z)/(z-z₀) dz
- Name
Cauchy's Integral Formula for Derivatives
- Note
For the n-th derivative of f(z) at z₀.
- Expression
f^(n)(z₀) = n!/(2πi) ∮C f(z)/(z-z₀)^(n+1) dz
- Name
Residue at a Simple Pole
- Note
For a simple pole z₀ of f(z).
- Expression
Res(f, z₀) = lim(z→z₀) [(z-z₀)f(z)]
- Name
Residue at a Pole of Order m
- Note
For a pole of order m at z₀.
- Expression
Res(f, z₀) = 1/((m-1)!) * lim(z→z₀) [d^(m-1)/dz^(m-1) ((z-z₀)ᵐ f(z))]
- Name
Residue Theorem
- Note
Σ(Residues inside C) is the sum of residues of f(z) at all its isolated singularities inside the contour C.
- Expression
∮C f(z) dz = 2πi * Σ(Residues inside C)
Prerequisites
Basic calculus (differentiation, integration, limits).
Knowledge of real series (Taylor series, Maclaurin series).
Vector calculus (partial derivatives).
Basic algebra of complex numbers (addition, subtraction, multiplication, division, modulus, argument).
Common mistakes
Incorrectly applying Cauchy-Riemann equations (e.g., mixing partial derivatives).
Errors in identifying the type of singularity (removable, pole, essential).
Incorrectly calculating residues, especially for poles of order m > 1.
Not checking for analyticity within and on the contour before applying Cauchy's Integral Theorem/Formula.
Sign errors when changing integration direction or using partial fractions.
Keywords
Complex numbers
Analytic functions
Cauchy-Riemann
Complex integration
Contour integral
Taylor series
Laurent series
Singularities
Poles
Residues
Residue Theorem
Conformal mapping
Practice preview
Calculate the residue of f(z) = 1/((z-1)(z-2)) at z = 1.…
medium
Find the modulus and principal argument of the complex number z = 1 + i*sqrt(3).…
easy
Which of the following functions satisfies the Cauchy-Riemann equations at all points?…
easy