Complex analysis deals with functions of a complex variable, especially derivatives and integrals of such functions. Holomorphic funktions, i.e. functions that are differentiable in a complex sense, have a number of interesting properties and applications. It turns out that the so-called imaginary number is very useful in investigating phenomena in our real reality. Information about the course round spring 2020 is available on canvas: https://kth.instructure.com/courses/17738

### Choose semester and course offering

Choose semester and course offering to see information from the correct course syllabus and course offering.

## Content and learning outcomes

### Course contents

Complex numbers in rectangular and polar form. Basic geometry and topology the the complex plae and on the Riemann sphere. Holomorphic, meromorphic and harmonic functions. Conformal mappings. Taylor and Laurent series. Radius of convergence and termwise differentiation and integration of power series. Classification of singularities. Poles and zeros, the argument principle and Rouchés theorem. Liouvilles theorem with applications. Differentiation and integration in the complex plane. Cauchy- Riemann equations. Cauchys theorem and Cauchy's integral formula with corollaries. The maximum principle. Residues. Applications to, for example, transform theory, heat conduction and electricity theory.

### Intended learning outcomes

After the course the student should be able to

• explain the meaning of basic concepts, theorems and methods within the parts of complex analysis described by the course content
• use concepts. theorems and methods to solve and present solutions to problems within the parts of complex analysis described by the course content,

in order to solve applied problems and to communicate with the help of mathematical language, even in other contexts.

• explain how different theorems and concepts are connected and deduce relationships from the given theorems.

### Course Disposition

No information inserted

## Literature and preparations

### Specific prerequisites

Completed basic course SF1626 Calculus in Several Variable or SF1674 Multivariable Calculus.

### Recommended prerequisites

No information inserted

### Equipment

No information inserted

### Literature

No information inserted

## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

A, B, C, D, E, FX, F

### Examination

• TEN1 - Exam, 7,5 hp, betygsskala: A, B, C, D, E, FX, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

The examiner decides, in consultation with KTHs Coordinator of students with disabilities (Funka), about any customized examination for students with documented, lasting disability. The examiner may allow another form of examination for re-examination of individual students.

### Opportunity to complete the requirements via supplementary examination

No information inserted

### Opportunity to raise an approved grade via renewed examination

No information inserted

### Ethical approach

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

## Further information

### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web SF1691

SCI/Mathematics

Technology

First cycle