URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=2DME30&collegejaar=2026&taal=enDescription
Recap of computation with complex numbers.Convergence of sequences, series and power series.
Complex functions: limits, continuity and differentiability, holomorphic functions and the Cauchy-Riemann equation, representation of holomorphic functions by power series.
Integration of complex functions: curves and contours, integration along a curve, Cauchy’s integral theorem, residues, Cauchy’s integral formula and Taylor series, entire functions and Liouville’s theorem, Laurent series and singularities, applications to the complex logarithm and the complex square root functions, using branch cuts.
Residue calculus: computation of residues, application to integration, integrals over a finite interval and over the real or complex axis, Jordan’s
lemma, applications to Fourier series and (inverse) Fourier- and Laplace transformation.
Objectives
- Being fluent in carrying out basic computations on complex numbers- Being able to check the convergence properties of a sequence of complex numbers and of a series of complex numbers, by applying several convergence tests (comparison tests and ratio- and root tests). Being able to find the radius of convergence of a power series using the ratio or root test.
- Having a good understanding of the notion of convergence for complex functions, including limits of functions and the definitions of continuity and differentibility for complex functions. Being able to check whether a complex function is holomorphic in a point or on an open set, using the Cauchy-Riemann relations. Being able to reconstruct a holomorphic function on the basis of its real or imaginary part, using the Cauchy-Riemann equation. Knowing that a power series is holomorphic inside its radius of convergence.
- Being able to compute the integral of a complex function along a curve, or to find an upper bound for its absolute value using the ML-lemma.
- Being able to apply Cauchy’s integral theorem for the computation of integrals along Jordan curves and its relationship to the definition of a residue in a singular point. Being able to determine residues of elementary functions. Understanding Cauchy’s integral formula, and Taylor’s theorem, i.e. the relationship between the coefficients of a Taylor series expansion, and the integrals of related functions. Being able to use these results to compute integrals using Taylor series expansions.
- Knowledge of Liouville’s Theorem, both in standard and generalized form, and being able to apply this result to reconstruct complex functions based on information of their growth, and the location and character of their singularities and zeros.
- Being able to apply Laurent’s theorem for the determination of the coefficients of a Laurent series of a given complex function, around a given (singular) point. Being able to determine the character of a singularity and its relationship with the principal part of the corresponding Laurent series.
- Being able to work with logarithmic functions and square root functions for complex numbers, and to use the notion of branch cuts.
- Being able to compute residues in isolated singular points, not only for simple poles, but also for poles of finite higher order, and in simple cases for essential singularities.
- Being able to apply residue calculus to determine finite integrals of functions defined in terms of sine and cosine functions, and use this for the determination of the Fourier coefficients of a periodic function. Knowledge of different types of convergence for Fourier series, and being able to recognize which convergence properties apply .
- Being able to apply residue calculus for integration over the real or imaginary axis, in particular for integrals that are needed to obtain (inverse) Fourier or Laplace transforms. Being able to apply Jordan’s lemma in this context.