MA-621 Complex Variable Theory

Emnet er tilknyttet følgende studieprogram

• Ph.d.-program i teknologi og realfag

Innhold

The course will present the following topics in complex variable theory, with additional topics and examples presented at the lecturer¿s discretion.

 

• Functions of one complex variable

• Cauchy-Riemann equations

• Singularities and their classification

• Riemann surfaces and branch cuts

• Residues and contour integration

• Techniques for closing the contour in the complex plane

• Examples of special contours

• Bromwich integral for inversion of Laplace transforms

• Summation of series with contour integration

• Method of steepest descents

• Conformal mapping

• Solving odes in the complex plane

• Bessel functions, Legendre functions, hypergeometric functions and orthogonal polynomials

• Examples such as Green functions and dispersion relations

• Possible selected advanced topics

Læringsutbytte

The course will introduce students to the standard results and techniques in complex variable theory which are necessary for research in Applied Mathematics. The course will present the material in the applications-oriented manner commonly employed in Applied Mathematics and Physics. The successful student will become familiar with contour integration using the method of residues and will be able to apply this technique to obtain the Green functions of standard partial differential equations and to invert Laplace transforms. The student will also be familiar with conformal mapping techniques, the solution of ordinary differential equations in the complex plane and will also be familiar with some common special functions.

Undervisnings- og læringsformer

Lectures + Examples Classes

Eksamen

Oral Examination. Pass/Fail