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XVI.–On the Theory of Continued-Fractions

Published online by Cambridge University Press:  15 September 2014

E. T. Whittaker
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Extract

The value of the continued-fraction, as a means of representing an analytic function f(x), is now fully recognised. Other representations, such as power-series or Fourier series or Dirichlet series or series of inverse factorials, converge in general only over limited regions of the x-plane, and fail to converge over the rest of the plane, whereas the representation of the function by a continued-fraction converges (in a large class of cases at any rate) over the whole x-plane, with the exception of certain singular curves.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 36 , Issue 3-4 , 1917 , pp. 243 - 255
Copyright
Copyright © Royal Society of Edinburgh 1917

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References

page 247 note * Phil. Mag. (4), 5, p. 446, 6, p. 297 (1853); Math. Papers, 1, pp. 609, 641.

page 247 note † Phil. Trans., 148 (1858), p. 17; Coll. Papers, 2, p. 475.

page 247 note ‡ Berlin Monatsb., 1873, p. 117; Berlin Sitzungsb., 1890, p. 1081.

page 247 note § Annales de la Fac. des Sc. de Toulouse, 3 (1889), H.

page 248 note * This process, which is due to Euler, Nova Acta Petrop., 2 (1784), p. 36, consists in the repeated use of the identity

page 249 note * This latter condition excludes the comparatively worthless method of converting infinite series into continued-fractions which is commonly given in elementary text-books on algebra, and which may be expressed by the equation

for in this equation the n th convergent to the continued-fraction coincides with the series only so far as its n th term, instead of as far as its 2n th term.

page 250 note Journal für Math., 33 (1845), p. 174.

page 252 note * If D = | a11, a22, a33, … ann | is any determinant, and if Ars denotes the co-factor of a rs in D, then the determinant | A11, A22, A33, … Ann | is called the adjugate of D. This nomenclature follows that of Cauchy's original memoir, and is always used by Sir Thomas Muir in his extensive writings on determinants: some writers have improperly used the word reciprocal in the sense of adjugate. We use the word reciprocal to signify the determinant which is the determinant of the matrix reciprocal to the matrix (a11, a22, … ann).