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Hypercomplex Analysis - NMAG461

Title:	Hyperkomplexní analýza
Guaranteed by:	Mathematical Institute of Charles University (32-MUUK)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2023
Semester:	winter
E-Credits:	3
Hours per week, examination:	winter s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech, English
Teaching methods:	full-time
Teaching methods:	full-time

Guarantor:	doc. RNDr. Roman Lávička, Ph.D.
Class:	M Mgr. MSTR M Mgr. MSTR > Volitelné
Classification:	Mathematics > Geometry, Real and Complex Analysis
Is interchangeable with:	NMAA039

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Annotation -

Last update: T_MUUK (16.05.2013)

Clifford algebras, Dirac equation, properties of solutions of the Dirac equation on Rn. Monogenic functions, Cauchy theorem and Cauchy integral formula. Taylor and Laurent series. Residuum of monogenic functions, residuum theorem. Conformal invariance.

Course completion requirements -

Last update: doc. RNDr. Roman Lávička, Ph.D. (23.06.2021)

Students should pass an examination.

Literature - Czech

Last update: doc. RNDr. Roman Lávička, Ph.D. (11.10.2016)

[1] F. Brackx, R. Delanghe, F. Sommen: Clifford analysis, Pitman, London, 1982.

[2] R. Delanghe, F. Sommen, V. Souček, Clifford algebra and spinor- valued functions, Kluwer Academic Publishers, Dordrecht, 1992.

[3] J.E. Gilbert, M.A.M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, Cambridge, 1991.

[4] P. Lounesto, Clifford Algebras and Spinors, Cambridge University Press, 1997 (Second edition, 2001).

Requirements to the exam -

Last update: doc. RNDr. Roman Lávička, Ph.D. (23.06.2021)

Requirements to the exam correspond to the syllabus to the extent to which topics were covered during the course.

Syllabus -

Last update: doc. RNDr. Roman Lávička, Ph.D. (13.09.2013)

Clifford algebra of a vector space with a scalar product, Spin group, spinor representations, spinor fields, the Dirac operator. Monogenic functions, connection with harmonic functions, Cauchy theorem, Cauchy integral formula and its applications, Morera theorem, Taylor and Laurent series, residue, residue theorem. Conformal maps in Euclidean space, action of conformal transformations on monogenic functions, conformal invariance.