1915-16.] On the Theory of Continued-Fractions. 
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XVI. — On the Theory of Continued-Fractions. 
By Professor E. T. Whittaker. 
(MS. received May 18, 1916. Read June 5, 1916.) 
I. Introduction. 
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 cc-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 &-plane, with the exception of certain singular 
curves. For example, the power-series 
1 1 1.3 1.3.5 
2* + 2 ! 2 2 * s+ 3 ! 2V + 4! 2V + ' ' ' 
converges only in the region which lies outside a circle of radius unity 
whose centre is the origin, although the function which it represents, 
namely, {x — (x 2 ~ 1)*}, exists inside that circle; but the continued-fraction 
representation of the same function, namely, 
Yx~— i 
2 *-2z 
2x 
converges over the whole x-plane, with the exception of the part of the 
real axis of x between x— — 1 and x = +1. 
The reason for the superiority of continued-fractions over power-series 
is that in a continued-fraction the numerators and denominators of the 
successive convergents are each helping on the approximation, whereas a 
power-series consists (so to speak) of a numerator alone, and being thus 
restricted, is unable to provide an approximation which is either so rapid 
or so widely applicable as that provided by the continued-fraction. 
The great impediment to the use of continued-fractions in Theory of 
Functions and Differential Equations is the want of algorithms for adding, 
multiplying, and differentiating them. The object of the present paper is 
to supply in some measure this deficiency. 
