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Published online by Cambridge University Press: 15 September 2014
It is well known that, if the infinite series
is convergent for any non-integral value of z, it is uniformly convergent in any finite region of the z-plane and represents an integral function C(z), say, such that C(n) = an for n = 0, ± 1, ± 2, … It is called the Cardinal Function of the table of values, and is identical with Gauss's Interpolation Formula (suitably bracketed).
The function C(z) defined by the series (1) has been given two different definite integral representations, due to Ferrar and Ogura respectively.
page 220 note * Whittaker, E. T., Proc. Roy. Soc. Edin., 35 (1915), 181–194.CrossRefGoogle Scholar
page 220 note † Ferrar, W. L., Proc. Roy. Soc. Edin., 45 (1924), 269–282 (273).CrossRefGoogle ScholarOgura, K., Tûhoku Math. Journ., 17 (1920), 232–241 (240).Google Scholar
page 220 note † Whittaker, J. M., Proc. Edin. Math. Soc. (2), 1 (1927–1929), 41–46CrossRefGoogle Scholar; Theorem I.
page 223 note * Cf. Ferrar, W. L., Proc. Roy. Soc. Edin., 47 (1927), 230–242 (238).CrossRefGoogle Scholar
page 223 note † Hobson, , Functions of a Real Variable, 2 (1926), 552.Google Scholar
page 224 note * Added on 27th June 1930, at the suggestion of a referee.
page 224 note † Proc. Lond. Math. Soc. (2), 25 (1926), 283–302.
page 224 note ‡ Quarterly Journal (Oxford), 1 (1930), 38–59.