Geomathematically oriented potential theory by Willi Freeden, Christian Gerhards

By Willi Freeden, Christian Gerhards

''This paintings explores and offers the rules of 'surface capability idea for the sphere,' as well as these within the Euclidean house R3, and breaks new mathematical flooring in dealing generically with power theoretic points of gravitation and geomagnetism. The paintings covers a two-semester graduate direction within the educating cycle of geomathematics, however it can to boot be used as a reference for researchers and Read more...

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3 Vector Inner/Outer Harmonics . . . . . . . . . . . . . . . 6 Tensor Spherical Harmonics . . . . . . . . . . . . . . . . . . . . 1 Radial-Tangential System . . . . . . . . . . . . . . . . . . 2 Eigenfunction System . . . . . . . . . . . . . . . . . . . . 3 Tensor Outer Harmonics . . . . . . . . . . . . . . . . . . 7 Exercises . . . . . . . . . . . . . . . . . . . .

4 Inner/Outer Harmonics . . . . . . . . . . . . . . . . . . . 4 (Scalar) Circular Harmonics . . . . . . . . . . . . . . . . . . . . 1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . 2 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . 3 Stereographic Projection . . . . . . . . . . . . . . . . . . 4 Inner/Outer Harmonics . . . . . . . . . . . . . . . . . . . 5 Vector Spherical Harmonics .

Freeden, T. Gervens, M. Schreiner [1998], W. Freeden, M. N. Lebedev [1973], C. M¨ uller [1966], C. M¨ uller [1998], and G. Szeg¨ o [1939]). Further properties can also be found in the exercises. 14. A polynomial Pn : [−1, 1] → R of degree n ∈ N0 is called a Legendre polynomial (of degree n) if the following properties are satisfied: (i) 1 −1 Pn (t)Pm (t)dt = 0, n = m, (ii) Pn (1) = 1. 14. They have the explicit representation n 2 Pn (t) = (−1)k k=0 (2n − 2k)! k! t ∈ [−1, 1]. 2. , d d (1 − t2 ) dt dt Pn (t) = −n(n + 1)Pn (t), t ∈ [−1, 1].

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