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Syllabus:

# Complex Analysis, 7.5 Credits

Swedish name: Komplex analys

This syllabus is valid: 2024-01-15 and until further notice

Course code: 5MA077

Credit points: 7.5

Education level: First cycle

Main Field of Study and progress level: Mathematics: First cycle, has at least 60 credits in first-cycle course/s as entry requirements

Responsible department: Department of Mathematics and Mathematical Statistics

Revised by: Faculty Board of Science and Technology, 2023-11-06

## Contents

The course is an introduction to complex analysis in one variable. The course deals with 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. The course also covers one or more applications of the theory are reviewed.

## Expected learning outcomes

For a passing grade, the student must be able to:

• explain the concepts of analytic functions and harmonic function and the importance of the Cauchy Riemann equations
• explain and apply Cauchy's integral formula and some of its consequences
• explain the convergence of power series and develop analytical capabilities in Taylor or Laurent series in a given domain
• describe the basic properties of singularities and zeros of analytic functions and calculate residues and use these to calculate integrals
• account for conformal mappings and its connection with analytic functions

## 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 i assigned: Fail (U), Pass (G), Pass with distinction (VG). A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for two years following the student's first participation in 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 adressed 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.umu.se, 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.

## Literature

### Valid from: 2024 week 3

Fundamentals of complex analysis : engineering, science, and mathematics
Saff Edward B., Snider Arthur David
3. ed. : Harlow [England] : Pearson : c2014 : 516 s. :
ISBN: 9781292023755 (pbk.)
Mandatory
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