Complex Variables
A rigorous first course in complex analysis for mathematics and physics students: the algebra and geometry of complex numbers, polar form and roots of unity, functions as mappings of the plane and the Riemann sphere, limits and complex differentiability, the Cauchy–Riemann equations and harmonic functions, the exponential, the multivalued logarithm, complex powers and branch cuts, contour integrals, winding numbers and the ML inequality, Cauchy's theorem with Goursat's proof and contour deformation, Cauchy's integral formula, Liouville's theorem and the fundamental theorem of algebra, Taylor and Laurent series, singularities, residues, the residue theorem and the argument principle, the evaluation of real integrals by residues, and a capstone on Möbius transformations, the Riemann mapping theorem and the Joukowski airfoil, with complete proofs and runnable Python examples throughout.
MATH-334 Year 3 3 credits 12 lessons · 5 h 2 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- Perform arithmetic with complex numbers in Cartesian and polar form, use the modulus, argument and conjugate, apply de Moivre's formula and find all roots of a complex number.
- Describe sets in the complex plane, interpret functions of a complex variable as mappings, and find the images of lines, circles and regions under linear maps, powers, inversion and the exponential.
- Prove and disprove complex limits, use path dependence to show limits fail, and decide from the definition where a function is complex differentiable.
- Derive and apply the Cauchy–Riemann equations in Cartesian and polar form, prove the rigidity properties of analytic functions, and find harmonic conjugates.
- Work with the complex exponential, trigonometric and hyperbolic functions, the multivalued logarithm and complex powers, and choose branches and branch cuts suited to a region.
- Parametrize contours, compute contour integrals, use antiderivatives and winding numbers, and bound integrals with the ML inequality.
- State and prove Cauchy's theorem, including Goursat's argument for triangles, and use contour deformation to evaluate integrals on regions with holes.
- Apply Cauchy's integral formula and its derivative form, and prove Cauchy's estimates, Liouville's theorem, the fundamental theorem of algebra, Morera's theorem and the maximum modulus principle.
- Find radii of convergence, expand analytic functions in Taylor series, use the identity theorem, and compute Laurent series on annuli.
- Classify isolated singularities, compute residues at poles and essential singularities, and apply the residue theorem and the argument principle.
- Evaluate real integrals by residues using the unit circle, semicircular, indented, keyhole and rectangular contours, together with Jordan's lemma.
- Construct conformal maps, including Möbius transformations from three points and the Joukowski map, and use them to solve problems for harmonic functions and ideal flow.
Course content
12 lessons · 5 h 2 min total · final exam
- 1 Complex numbers: algebra, geometry and polar form 24 min · quiz
- 2 Functions of a complex variable and mappings 20 min · quiz
- 3 Limits, continuity and complex differentiability 24 min · quiz
- 4 Analytic functions and the Cauchy–Riemann equations 23 min · quiz
- 5 Elementary functions: exponential, logarithm, powers and branches 24 min · quiz
- 6 Contour integrals 28 min · quiz
- 7 Cauchy's theorem 26 min · quiz
- 8 Cauchy's integral formula and its consequences 26 min · quiz
- 9 Power series, Taylor series and Laurent series 24 min · quiz
- 10 Singularities and the residue theorem 28 min · quiz
- 11 Evaluating real integrals with residues 31 min · quiz
- 12 Capstone: conformal mapping and applications 24 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A rigorous first course in complex analysis for mathematics and physics students: the algebra and geometry of complex numbers, polar form and roots of unity, functions as mappings of the plane and the Riemann sphere, limits and complex differentiability, the Cauchy–Riemann equations and harmonic functions, the exponential, the multivalued logarithm, complex powers and branch cuts, contour integrals, winding numbers and the ML inequality, Cauchy's theorem with Goursat's proof and contour deformation, Cauchy's integral formula, Liouville's theorem and the fundamental theorem of algebra, Taylor and Laurent series, singularities, residues, the residue theorem and the argument principle, the evaluation of real integrals by residues, and a capstone on Möbius transformations, the Riemann mapping theorem and the Joukowski airfoil, with complete proofs and runnable Python examples throughout.
Part of the Science stream, typically taken in year 3.