After successful completion of the course, students are able to...

• ... in the theory of complex functions ...
• ... investigate the complex differentiability of a given function via the Cauchy--Riemann differential equations and calculate conjugate harmonic functions. 
• ... calculate complex integrals over a parametrized curve or via the antiderivative.
• ... identify and classify the pols of a complex function and calculate the residuum at a pole.
• ... solve complex integrals via the Residue Theorem.
• ... in the vector space theory of a system of functions ...
• ... calculate the orthogonal projection of a given function onto the linear span of a given system of functions. 
• ... calculate the coefficiets of the Fourier series of a given function.
• ... determine the limit of the Fourier series of a given function at a specified point using the Dirichlet Theorem.
• ... in the theory of integral-transformations...
• ... calculate the Laplace tranform of a given function, via the definition and the elementary properties (linearity, similarity, differential, integration, shift, ...) of the Laplace transform.
• ... dertermine the inverse of the Laplace transform of a function using the complex inversion-formula and the residue theorem.
• ... solve initial value problems using the Laplace transform.
• ... calculate the Fourier transform and inverse Fourier transform, via the definition and the elementary properties (linearity, similarity, differential, shift, ...) of the Fourier transform.
• ... in the theory of linear partial differential equations ...
• ... classify a given linear partial differential equation (order, coefficients, homogen or inhomogen, type,...).
• ... determine a general solution of a given linear partial differential equation of first order via the method of characteristics.
• ... determine a general solution of classical homgeneous linear partial differential equations  of second order (potential equation, heat equation, wave equation,...) by method of seperation of variables and fit the general solution to a given set of boundary conditions via application of the theory of Fourierseries.

Laplace and Fourier transform, Complex Analysis, Fourier series, Partial Differential Equations

Lecture with exercises

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Written

The lecture notes are available at Grafischen Zentrum. Lecture notes from the winter semester 2020 can still be used.

Calculus, ODEs, vector spaces