Revised by: Faculty Board of Science and Technology, 2025-06-10
Contents
The course is an introduction to complex analysis in one variable. The course covers complex numbers and topology of the complex plane, analytic and harmonic functions, Cauchy-Riemann equations, complex integration, Cauchy's integral formula, power series and Laurent series, zeros and singularities, residue theory with the Cauchy residue theorem, the argument principle and conformal mappings, in particular Möbius transformations. The course also treats one or more applications of the theory.
Expected learning outcomes
For a passing grade, the student must be able to:
Knowledge and understanding
account for the concepts of analytic functions and harmonic functions and for the importance of the Cauchy Riemann equations
account for and apply Cauchy's integral formula and some of its consequences
account for the convergence of power series and expand analytic functions in Taylor or Laurent series in a given domain
account for the basic properties of singularities and zeros of analytic functions, calculate residues and use residue techniques to compute integrals
account for conformal mappings and their connection with analytic functions, and use Möbius transformations
Required Knowledge
The course requires courses in Mathematics minimum 60 ECTS or least two years of university studies and in both cases a course in multivariable calculus, minimum 7,5 ECTS or equivalent. Proficiency in English equivalent to the level required for basic eligibility for higher studies. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.
Form of instruction
The teaching mainly consists of lectures and lessons.
Examination modes
The course is examined by written exams. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG).
Deviations from the syllabus examination form can be made for a student who has a decision on pedagogical support due to disability. Individual adaptation of the examination form shall be considered based on the student's needs. The examination form is adapted within the framework of the expected learning outcomes of the course syllabus. At the request of the student, the course coordinator, in consultation with the examiner, must promptly decide on the adapted examination form. The decision shall then be communicated to the student.
A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.
Credit transfer All students have the right to have their previous education or equivalent, and their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be addressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.ͯÑÕÊÓƵ, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications.
Other regulations
In the event that the syllabus ceases to apply or undergoes major changes, students are guaranteed at least three examinations (including the regular examination opportunity) according to the regulations in the syllabus that the student was originally registered on for a period of a maximum of two years from the time that the previous syllabus ceased to apply or that the course ended.
