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Special Functions of Mathematical (Geo-)Physics

Lecture (4 SWS): every Monday  10:45 - 11:30 in 44-465 and every Wednesday  8:15 - 9:45 in 48-538 (as noted in KIS), first lecture on 12.04.2010
Lecturer: Martin Gutting
The lecture will be given in English unless unanimously desired otherwise.

Tutorials: every Tuesday 11:45 - 13:15 in 49-506. First tutorial on 27.04.2010.

For successful participation (i.e. a "Schein") you need to obtain at least 50% of the total points in the exercises (you can work in groups of up to three). You should present at least 1 exercise during the tutorials and you need to attend most of the tutorials (at most 2 absences). There will be a letter box for "Special Functions" next to the entrance of building 48 where you can hand in your results.

News

If you want to attend the lecture, but have a scheduling conflict, please contact me.

Note: Extra lecture on Friday 02.07. at 15:30 in 48-538 (instead of Wednesday 09.07.).

Note the change of room on Wednesday 14.04.2010. The lecture takes place in 36-265.

Lecture Notes as PDF: lecture notes (12.07.2010)

Lecture handout on spherical vector fields as PDF: Handout (30.06.2010)
Lecture handout on the hydrogen atom (incl. figures) as PDF: Handout (30.06.2010)
Lecture handout on spherical harmonics (incl. figures) as PDF: Handout (23.06.2010)
Lecture handout on interpolation as PDF: Handout (16.06.2010)
Lecture handout on Hermite and Laguerre polynomials as PDF: Handout (09.06.2010)
Lecture handout on spherical notation as PDF: Handout (09.06.2010)
Lecture handout on applications of Legendre polynomials as PDF: Handout (07.06.2010)

Exercises for the tutorials

sheet 11
sheet 10
sheet 9
sheet 8
sheet 7
sheet 6
sheet 5
sheet 4
sheet 3
sheet 2
sheet 1

Contents

The lecture gives an elementary approach to the theory of special functions in mathematical physics with special emphasis on geophysically relevant aspects. The essential topics of the lecture are in chronological order: the Gamma function, orthogonal polynomials, spherical polynomials (scalar, vectorial, and tensorial case), and Bessel functions. All fields will be assisted by geophysically relevant applications.

The lecture is a good preparation for further activities in the field of geomathematics such as "Constructive Approximation", "Potential Theory", "Inverse Problems", etc.

Inhalt

Die Vorlesung liefert einen elementaren Zugang zur Theorie der Speziellen Funktionen der Mathematischen Physik unter besonderer Berücksichtigung (geo-)physikalischer Anwendungen. Die wesentlichen Themen der Vorlesung sind in chronologischer Abfolge: die Gamma Function, orthogonale Polynome, sphärische Polynome (skalarer, vektorieller und tensorieller Fall) sowie Bessel Funktionen. Alle Themenbereiche werden begleitet durch geophysikalisch relevante Anwendungen.

Die Vorlesung ist eine gute Vorbereitung auf weitergehende Aktivitäten/Vorlesungen im Bereich Geomathematik wie zum Beispiel "Constructive Approximation", "Potential Theory", "Inverse Problems", etc.

Overview

  1. Introduction and Motivation.
  2. The Gamma Function.
  3. Orthogonal Polynomials.
  4. Spherical Harmonics.
  5. Vector Spherical Harmonics.
  6. Bessel Functions.

Literature

  1. W. Freeden, T.Gervens, M. Schreiner: Constructive approximation on the sphere. Clarendon Press, 1998.
  2. W. Freeden, V. Michel: Multiscale potential theory. Birkhäuser, 2004.
  3. W. Freeden, M. Schreiner: Spherical Functions of Mathematical Geosciences. Springer, 2009.
  4. N.N. Lebedew: Spezielle Funktionen und ihre Anwendungen. BI, 1973.
  5. C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17, Springer, 1966.
  6. C. Müller, Analysis of Spherical Symmetrics in Euclidean Spaces, Springer, 1998.
  7. G.P. Szegö: Orthogonal polynomials. American Math. Soc., 1967.
Further references are given in the lecture notes.