By Wolfgang Nolting

Der Grundkurs Theoretische Physik deckt in sieben Bänden alle für Diplom- und Bachelor/Master-Studiengänge maßgeblichen Gebiete ab. Jeder Band vermittelt das im jeweiligen Semester nötige theoretisch-physikalische Rüstzeug. Übungsaufgaben mit ausführlichen Lösungen dienen der Vertiefung des Stoffs. Band 1 behandelt die klassische Mechanik. Vorausgesetzt wird nur die übliche Schulmathematik, andere mathematische Hilfsmittel werden zu Beginn ausführlich erläutert. Die zweifarbig gestaltete Neuauflage wurde grundlegend überarbeitet und ergänzt.

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V > -1. )TisaneigenvectorofA, witha, = 2 - ' ( ( v + n ) ( v + n 1))-1/2, and bn = 0. Here c is a constant to make the eigenvector have norm = 1. 8) implies The right-hand side is The second sum is a telescoping sum whose value is zero. 2. 17) + + Proof. 16) follows. 17). 17) follows by integration and using j - 1 / 2 , k = ( k - l/2)7r. 3. If is positive semi-definite (definite) then the eigenvalues of T, are increasing (strictly increasing) functions of v. Proof. Apply the Hellmann-Feynman theorem.

Amer. Math. SOC115 (1992), 131-140. 31. M. E. H. Ismail and D. R. Masson, q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals, Trans. Amer. Math. 346 (1994), 63-116. 32. M. E. H. Ismail and M. E. Muldoon, On the variation with respect to a parameter of zeros of Bessel functions and q-Bessel functions, J. Math. Anal. 42 Appl. 135 (1988), 187-207. 33. M. E. H. Ismail and M. E. Muldoon, A discrete approach to monotonicity of zeros of orthogonal polynomials, Trans. Amer. Math.

PnPm,n(t) . + dm,n. 9) so it seems that {F,} are orthogonal with respect t o p. At t 4 co our pm,n(t) cannot have exponential growth, hence x E (0, co),and we obtain pm,n(t) = 5'm LW e-"tFm(z)Fn(x)dp(x). 10). It turns out the p must be a positive, hence {F,} are orthogonal with respect to p. 6) show that all zeros of F, and Qn lie in (0, co). 10) has been established by Karlin and McGregor in 35, 36. Later Karlin and McGregor 38 studied random walks on the state space of the nonnegative integers and defined another sequence of orthogonal polynomials.

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