Complex analysis with applications

The course is included in the study program physics and mathematics - bachelor. It may also be taken independent of study program.

This course presents the elementary theory of functions of a complex variable, including residue calculus and conformal mappings. It emphasizes the relationship between analytic functions and harmonic functions and the use of these concepts in solving boundary value problems, such as the two-dimensional Laplace equation. It is designed for students who need a foundational understanding of complex analysis in algebra, applied mathematics, physics, mathematical analysis, or statistics.

Applicants from Nordic countries:

Generell studiekompetanse og følgende spesielle opptakskrav: Matematikk R1 og i tillegg enten:

• Matematikk R2

• Fysikk 1 + 2 eller

• Kjemi 1+ 2 eller

• Biologi 1 + 2 eller

• Informasjonsteknologi 1 +2 eller

• Geofag 1 + 2 eller

• Teknologi og forskningslære 1 + 2

Recommended prerequisites is MAT-1020 Linear Algebra and MAT-1030 Vectorcalculus or equal.

Application code is 9197.

Knowledge

The student:

• understands the concept of an analytic function.

• understands the Riemann sphere and stereographic projection, and their role in extending the complex plane to include the point at infinity.

• understands the Cauchy-Riemann equations and their role in determining whether a function is analytic.

• understands the relationship between analyticity in a region and the ability to represent a function as a power series (e.g., Taylor or Laurent series).

• understands the concept of the radius of convergence and can determine it in simple cases.

• understands how the radius of convergence at a point relates to the domain of analyticity of a function.

• understands Liouville’s theorem and its application in proving the Fundamental Theorem of Algebra.

• understands Cauchy’s integral theorem and its consequences, including Cauchy’s integral formulas.

• understands the connection between analytic functions and potential theory (solutions of Laplace’s equation) in two dimensions.

• understands the properties of harmonic functions and their applications, for example to steady-state temperature distributions.

• understands the Argument Principle and its applications.

• understands conformal mappings and the Riemann Mapping Theorem.

• is familiar with the basic topology of planar domains, including the concepts of connectedness, simple connectivity, and multiple connectivity.

• understands the distinction between pointwise and uniform convergence of sequences and series of functions, and their implications for continuity, differentiability, and integrability.

• understands the differences between power series over the real and complex numbers, particularly the stronger convergence and analyticity properties in the complex case.

• is familiar with common analytic functions (polynomials, rational functions, exponential, trigonometric and hyperbolic functions, logarithmic functions, and power functions) and can work with combinations of these.

• is familiar with multivalued functions and their definition via branches and branch cuts in appropriate domains.

Skills

The student:

• can work confidently with complex numbers and functions of a complex variable.

• can classify isolated singularities using Laurent series in a punctured neighborhood of a singular point, including at infinity.

• can compute residues of functions at isolated singularities in standard cases.

• can apply Cauchy’s integral theorem and integral formulas in typical residue calculations.

• can construct explicit conformal mappings using Möbius transformations and (in standard cases) Schwarz-Christoffel transformations.

• can apply conformal mapping techniques to solve boundary value problems (e.g., for the 2D Laplace equation) arising in electrostatics, heat flow, and fluid mechanics.

General Competence

The student:

• can apply theoretical concepts from complex analysis to problems in mathematics, physics, and engineering.

• can independently analyse and solve problems involving analytic functions, singularities, and conformal mappings.

• demonstrates a systematic understanding of the interplay between theory (e.g., Liouville’s theorem, Cauchy’s theorem) and applications (e.g., boundary value problems and differential equations).

• can communicate mathematical reasoning and solutions clearly and effectively in the context of complex analysis.

Anbefalte forkunnskaper

MAT-1020 Linear algebra, MAT-1030 Vector Calculus

Obligatorisk forkunnskapskrav

MAT-1010 Calculus

If you pass the examination in this course, you will get an reduction in credits (as stated below), if you previously have passed the following courses:

• MA-214 Elementary complex analysis 9 ects
• MAT-2100 Complex Analysis 10 ects

Language of instruction and examination

Schedule

This course is open to incoming exchange students.

Study Level: Bachelor's

Admission prerequisites: To take this course, you must first meet the requirements listed in the "Admission requirements" section above.

For details on how to apply for exchange, course selection guidelines, or to contact the Incoming Admissions Team, please visit: Admission for Student Exchange.

More info about the coursework requirements

• 6 quizzes
• 1 physical midterm exam
• up to 3 mandatory submissions

Approved mandatory assignments are valid only for the current semester and for the re-sit exam.

Re-sit examination