Course syllabus - Introduction to Real and Complex Analysis

Objectives

The course aims at giving the student a fundamental knowledge, tools and methods on the topic of real and complex analysis in one variable. The course also aims at giving a deeper theoretical understanding for mathematics, but even more how the theory may be applied in science and technology.

Learning outcomes

At the end of a passed course, the student is expected to be able to

- define and apply such topological concepts as open, closed, bounded, compact set respectively, and boundary of a domain

- utilize the definition of a limit for simple evaluations of limits, to be able to check if a function is continuous at a point, and also to have a conceptual understanding of continuous curves and simply connected regions

- apply different forms of representations of complex numbers, and be able to apply the arithmetical operations and triangle inequality for real and complex numbers

- solve fundamental equations in one complex variable

- map regions by using the Möbius transformation

- evaluate line integrals of elementary complex valued functions, and to be able to determine whether the integral is dependent of path

- determine whether a function is complex differentiable or not, and also to be able to determine whether a function satisfies the Cauchy-Riemann equations or not, i.e. to be able to determine whether the function is analytical

- find the power series of an analytic function, to be able to interpret Laurent series, and to be able to do evaluations by residue calculus, even applied to real-valued integrals

Course content

- Set topology: interior points, open set, closed set, bounded set, compact set, region, connected region, simply connected region, boundary

- Limits: formal definition, continuity, continuous curves

- Complex numbers: real part, imaginary part, polar form, normal form. Absolute value, triangle inequality and de Moivre’s formula

- Solving equations: quadratic functions and power equations

- The Möbius mapping: Möbius transformations

- Differentiability: complex differentiable function, Cauchy-Riemann equations

- Elementary functions: power functions, polynomials, rational functions, the exponential function and trigonometric functions

- Line integrals: integration of complex-valued functions on parameter form, line integrals, primitive function

- Power series and Laurent series: zero of order n, isolated singularities, removable singularity, pole of order n, essential singularity, generalized real integrals

- Theorems presented are among others: Heine-Borel theorem, a theorem about the equivalence of differentiability, analytic function and Cauchy-Riemann equations, a theorem about the path-dependence of the line integral, identity theorem for analytic functions, Cauchy integral theorem, Cauchy integral formula, a theorem about isolated zeroes for analytic functions, a theorem about analytic functions defined in an annulus, residue theorem

Tuition

Teaching is given in the form of lectures and classes.

Specific requirements

At least totally 60 credits in the technical, natural sciences, business administration or economics areas including Basic Vector Algebra, 7.5 credits, Single Variable Calculus, 7.5 credits, of which 6 credits must be completed at the beginning of the course, and Calculus of Several Variables 7.5 credits, of which 2.5 credits must be completed at the beginning of the course, or the equivalent.

Examination

Assigned problems (INL1) , 2,5 credits, marks Fail (U), Pass (G)
Written and/or oral examination  (TEN1), 5 credits, marks Fail (U), 3, 4, 5

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Grading scale: 5, 4, 3