Functions of complex variables, contour integration, and analytic functions
← Back to MathematicsMaster the foundations of complex numbers and their geometric interpretation.
Learn the fundamental concept of analyticity and complex differentiability.
Study complex versions of familiar functions and their properties.
Master contour integration and its fundamental theorems.
Study convergence of complex series and power series expansions.
Learn the powerful residue theorem for evaluating integrals.
Study angle-preserving transformations and their applications.
Explore important special functions in complex analysis.
Explore advanced topics and applications of complex analysis.
Connect complex analysis to other areas of mathematics and physics.
Build the foundation with complex number arithmetic and geometric visualization.
Master operations with complex numbers z = a + bi. Learn addition, multiplication, division, and the properties that make ℂ a field extension of ℝ.
Convert between z = a + bi and z = r(cos θ + i sin θ) = re^(iθ). Understand when each form is most useful for computations.
Apply (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) to compute powers and roots. Understand the geometric interpretation of multiplication as rotation and scaling.
Find all nth roots of complex numbers. Understand how the n roots are evenly distributed around a circle in the complex plane.
Visualize complex numbers as points in the Argand diagram. Learn how complex operations correspond to geometric transformations.
Understand the mapping between the complex plane and the Riemann sphere. Learn how this provides a geometric model for the extended complex plane.
Study ℂ ∪ {∞} and understand the point at infinity. Learn how this compactification is essential for certain theorems and applications.
Understand |z₁ - z₂| as distance, arg(z₁/z₂) as angle between vectors, and how complex multiplication combines rotation and scaling.
Learn the fundamental concept that distinguishes complex analysis from real analysis.
Define f'(z₀) = lim[z→z₀] (f(z) - f(z₀))/(z - z₀). Understand why this limit must be independent of the direction of approach in ℂ.
Learn the necessary conditions ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x for f = u + iv to be analytic. Understand their geometric meaning.
Understand that complex differentiability at a point vs. in a neighborhood leads to the equivalence of "analytic" and "holomorphic" functions.
Study functions satisfying Laplace's equation ∇²u = 0. Learn the connection between harmonic functions and real/imaginary parts of analytic functions.
For harmonic function u, find harmonic conjugate v such that f = u + iv is analytic. Understand uniqueness up to constants.
Learn that analytic functions with non-zero derivative preserve angles. Understand the geometric significance of conformality.
Study polynomials, exponential, and other elementary functions. Learn to verify analyticity using Cauchy-Riemann equations.
Identify points where functions fail to be analytic. Learn about removable singularities, poles, essential singularities, and branch points.
Extend familiar real functions to the complex domain and understand their new properties.
Define e^z = e^x(cos y + i sin y) for z = x + iy. Learn Euler's formula e^(iθ) = cos θ + i sin θ and the periodicity e^(z+2πi) = e^z.
Define log z as the inverse of e^z. Understand multivaluedness and the need for branch cuts. Learn about principal branches and logarithmic branches.
Define sin z = (e^(iz) - e^(-iz))/(2i) and cos z = (e^(iz) + e^(-iz))/2. Learn how these extend real trigonometric functions and their new properties.
Study sinh z, cosh z, and tanh z in the complex plane. Understand their relationship to trigonometric functions and periodicity properties.
Define z^w = e^(w log z) for complex z and w. Understand multivaluedness and how to choose appropriate branches for applications.
Study inverse trigonometric and hyperbolic functions in ℂ. Learn about branch cuts and principal values for multivalued inverse functions.
Understand how to make multivalued functions single-valued by introducing branch cuts. Learn standard conventions for principal branches.
Visualize multivalued functions on Riemann surfaces where each "sheet" corresponds to a different branch. Understand how this resolves multivaluedness.
Master contour integration and prove the fundamental theorems of complex analysis.
Define ∫_C f(z) dz for complex functions along curves C. Learn how to parameterize curves and compute integrals directly.
Study piecewise smooth curves, simple closed curves, and their orientation. Understand how curve direction affects integral values.
Prove that ∮_C f(z) dz = 0 for analytic f on simply connected domains. Understand this as the fundamental theorem of complex analysis.
Learn f(z₀) = (1/2πi) ∮_C f(z)/(z-z₀) dz. Understand how this expresses analytic functions in terms of their boundary values.
Prove f^(n)(z₀) = (n!/2πi) ∮_C f(z)/(z-z₀)^(n+1) dz. Understand how this shows infinite differentiability of analytic functions.
Prove that |f| attains its maximum on the boundary of domains for non-constant analytic functions. Learn applications to uniqueness theorems.
Prove that bounded entire functions are constant. Understand this as a fundamental rigidity result in complex analysis.
Use Liouville's theorem to prove that every non-constant polynomial has a root. Understand this elegant complex-analytic proof.
Study convergence of complex series and power series expansions of analytic functions.
Extend convergence concepts to ℂ. Learn that convergence means |z_n - L| → 0, and study absolute vs. conditional convergence.
Study series of the form Σa_n(z-z₀)^n. Learn about convergence within circles and the radius of convergence formula.
Apply the Cauchy-Hadamard theorem: R = 1/limsup|a_n|^(1/n). Learn ratio and root tests for determining convergence radius.
Expand analytic functions as f(z) = Σf^(n)(z₀)(z-z₀)^n/n!. Understand that analytic functions are "locally polynomial."
Extend to series with negative powers: f(z) = Σa_n(z-z₀)^n for all integers n. Learn expansion in annular regions.
Extend analytic functions beyond their original domains using power series. Understand uniqueness and the identity theorem.
Study when series converge uniformly on compact sets. Learn Weierstrass M-test and how uniform convergence preserves analyticity.
Learn that uniform limits of analytic functions are analytic, and that series can be differentiated and integrated term by term.
Master the residue theorem and its applications to evaluating complex and real integrals.
Classify singularities using Laurent series: removable (no negative powers), poles (finite negative powers), essential (infinite negative powers).
Define Res(f,z₀) as the coefficient of (z-z₀)^(-1) in Laurent expansion. Learn ∮f(z)dz = 2πi Σ residues inside contour.
Distinguish removable singularities, poles of various orders, and essential singularities. Learn tests using limits and Laurent series.
Learn formulas for residues at simple poles: Res(f,z₀) = lim(z-z₀)f(z), and higher-order poles using derivatives.
Use residue theorem to evaluate ∫₋∞^∞ f(x)dx by integrating over semicircular contours and applying residue theorem.
Evaluate ∫₀^(2π) R(cos θ, sin θ)dθ by substituting z = e^(iθ) and integrating around the unit circle.
Handle integrals with branch cuts and multivalued functions. Learn keyhole contours and other specialized techniques.
Learn conditions for ∫_(semicircle) f(z)dz → 0 as radius → ∞. Essential for applying residue theorem to real integrals.
Study angle-preserving transformations and their applications to physics and engineering.
Learn that analytic functions with f'(z) ≠ 0 preserve angles and shapes locally. Understand the geometric significance of conformality.
Study f(z) = (az+b)/(cz+d) as the most general conformal maps of the extended plane. Learn about fixed points and classification.
Learn the invariant (z,w;u,v) = ((z-u)(w-v))/((z-v)(w-u)) preserved by linear fractional transformations. Apply to mapping problems.
Understand that any simply connected domain (except ℂ) can be mapped conformally to the unit disk. Learn about the significance and limitations.
Learn the explicit formula for mapping the upper half-plane to polygonal regions. Understand parameter determination and applications.
Apply conformal mappings to solve Laplace's equation in complex domains. Understand how conformality preserves harmonic functions.
Model 2D incompressible flow using complex velocity potentials. Learn how conformal mappings transform flow patterns around obstacles.
Solve electrostatic problems using conformal mappings. Understand how electric field lines and equipotentials transform under conformal maps.
Explore important special functions that arise naturally in complex analysis.
Study Γ(z) = ∫₀^∞ t^(z-1)e^(-t) dt for Re(z) > 0 and its analytic continuation. Learn functional equation Γ(z+1) = zΓ(z).
Learn B(z,w) = ∫₀¹ t^(z-1)(1-t)^(w-1) dt and its relation to gamma function: B(z,w) = Γ(z)Γ(w)/Γ(z+w).
Study ζ(s) = Σn^(-s) for Re(s) > 1 and its analytic continuation. Learn about zeros, poles, and connection to prime numbers.
Study doubly periodic functions on ℂ. Learn about fundamental domains, Weierstrass ℘-function, and modular transformations.
Learn Jacobi theta functions and their role in elliptic function theory. Understand transformation properties and applications.
Study solutions to Bessel's differential equation. Learn series representations, asymptotic behavior, and applications to physics.
Study ₂F₁(a,b;c;z) and its properties. Learn how many special functions are special cases of hypergeometric functions.
Learn to find asymptotic series for special functions. Understand the difference between convergent and asymptotic series.
Explore advanced theorems and techniques in complex function theory.
Learn (1/2πi)∮(f'/f)dz = N - P where N = zeros, P = poles inside contour (counting multiplicity). Apply to root counting.
If |g(z)| < |f(z)| on |z| = r, then f and f+g have the same number of zeros inside |z| = r. Learn applications to polynomial roots.
Deepen understanding of maximum modulus principle and its consequences: minimum modulus principle, open mapping theorem, and uniqueness results.
If f: D → D with f(0) = 0, then |f(z)| ≤ |z| and |f'(0)| ≤ 1. Learn the rigidity: equality characterizes automorphisms.
Extend maximum principle to unbounded domains using growth conditions. Learn applications to uniqueness problems in strips and sectors.
Study functions analytic on all of ℂ. Learn Liouville's theorem, Hadamard's factorization theorem, and order of growth classification.
Study functions with only poles as singularities. Learn partial fraction decompositions and the connection to rational functions.
Study families of analytic functions with uniform bounds. Learn Montel's theorem and applications to iteration theory and dynamics.
Connect complex analysis to other areas of mathematics, physics, and engineering.
Use complex analysis to understand transforms F(ω) = ∫f(t)e^(-iωt)dt. Learn about poles, residues, and inversion formulas.
Apply complex analysis to quantum mechanics: wave functions, scattering theory, and the role of analytic continuation in physics.
Learn how complex analysis proves theorems in number theory: prime number theorem, Dirichlet's theorem on primes in arithmetic progressions.
Understand connections to Riemann surfaces, algebraic curves, and how complex analysis provides geometric intuition for abstract algebra.
Apply complex methods to ODEs: Laplace transforms, series solutions, and asymptotic analysis of solutions in the complex plane.
Study applications to electromagnetism, fluid dynamics, heat conduction, and wave propagation using complex variable techniques.
Apply complex analysis to digital signal processing: z-transforms, filter design, and stability analysis using pole-zero plots.
Use complex analysis in control systems: transfer functions, stability criteria using Nyquist plots, and root-locus techniques.