Complex Analysis
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Faculteit  Science and Engineering 
Jaar  2018/19 
Vakcode  WICA07 
Vaknaam  Complex Analysis 
Niveau(s)  bachelor 
Voertaal  Engels 
Periode  semester I b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Complex Analysis  
Leerdoelen  1. The student is able to reproduce basic concepts, theorems, and proofs from complex analysis (such as analyticity of complex functions, the CauchyRiemann equations, the Cauchy theorem and the Cauchy integral formula, Taylor series and Laurent series of complex functions, the maximum modulus principle, the principle of the argument, etc.), and is able to apply them to obtain other related theoretical results. 2. The student is able to determine at which points a given complex function is analytic or differentiable, to compute the integrals of complex functions over given contours, and to calculate Taylor series and Laurent series for given complex functions. 3. The student is able to prove the properties of the elementary complex functions, and is able to use these properties to investigate the behavior of these functions and functions constructed through them. 4. The student is able to reproduce the definitions of the various types of singularities of complex functions, and is able to classify the singularities of given elementary complex functions. 5. The student is able to calculate the residues at singularities of given complex functions, and is able to use the calculus of residues in order to calculate the value of given improper integrals of real functions. 6. The student is able to reproduce the principle of the argument and the theorem of Rouché, and is able to use these to relate the properties of complex functions to the location of their zeros. 

Omschrijving  Complex analysis treats complex valued functions defined in the complex plane focusing on those functions that are analytic, i.e., differentiable. These include functions from calculus, such as exponential, sine, cosine, logarithm and square root, but also polynomials, quotients of polynomials , and functions which can be composed out of these. Complex analytic functions have nice intrinsic properties, which simplify their study and provide powerful techniques for the computation of integrals of both complex and real functions. In this direction, an important aspect of the course is the treatment of the calculus of residues, by means of which such integrals can be evaluated. Below a list of topics to be treated: • Complex functions, complex differentiability. • Analytic functions. • CauchyRiemann equations. • Harmonic functions. • Complex trigonometric functions, complex logarithm. • Path integrals, Cauchy theorem, Cauchy integral formula. • Maximum modulus principle. • Taylor series, Laurent series. • Singularities of complex functions. • Calculus of residues. • Principle of the argument, theorem of Rouché, winding number. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC), Werkcollege (T)  
Toetsvorm 
Opdracht (AST), Schriftelijk tentamen (WE)
(Assessment take place through homework assignments, mid term exam, written exam. The final grade is obtained by considering the average of the best 5 out of 6 of the homework assignments (HW), the grade of the midterm exam (ME), and the grade of the final exam (FE). Specifically, the final grade is the maximum of FE or 0.75FE+0.25ME or 0.75FE+0.25HW or 0.6FE+0.2ME+0.2HW.) 

Vaksoort  bachelor  
Coördinator  K. Efstathiou, PhD.  
Docent(en)  K. Efstathiou, PhD.  
Verplichte literatuur 


Entreevoorwaarden  
Opmerkingen  
Opgenomen in 
