By Kirkwood J.R.

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**Extra resources for An introduction to analysis**

**Example text**

35, 3034. Milne-Thompson, L. M. (1951). ” Macmillan, New York. Muir, T. “A Treatise on the theory of Determinants” (revised by W. H. Metzer), p. 372. Dover, New York. Nevanlinna, R. Ann. Acad. Sci. Fennicae, Ser. A 13, 1. Nuttall, J. Phys. Rev. 157, 1312. Private communication. Ann. Ecole Super. , 1. Peres, A. J. Math. Phys. 4, 332. ,and Henrici, P. Contemporary problems in the theory of analytic functions (in Russian). Int. Conf. A. Lavrent’ev, (ed). Izdat. “Nauka,” Moscow, 1966. Riesz, F. (1911).

44) in the form analogous to the Casimir-Polder expression. We recognize ,8(iy)of Eq. (45) as a series of Stieltjes, for which bounds can be constructed from the Pad6 approximants, given the appropriate moments (Chang and Karplus, 1970). From the power-series expansion with we see that the required moment coefficients, , 8 k , are distinct from the ,uk of Eq. (12). k cannot be determined directly from experiment. 5 - k)' which has been suggested by Dalgarno and Kingston (1960) and by Bell (1965) as a reliable approximation to the sum-rule values.

Thus ablas is nonpositive-definite. It goes to zero as a goes to infinity in order that b remain nonnegative. Similarly, a2b/as’ must go to zero. We can now repeat this argument for a2b/as2,a3b/as3,a4b/as4 and thus by induction all odd derivatives of b are nonpositive-definite. Consequently the inequalities imply the derivative condition as well as vice versa. One may inquire what class of functions is characterized by property (108). At least for functions which have certain regularity and boundedness conditions, this class of functions is easily characterized.