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XVIII.—On the Functions which are represented by the Expansions of the Interpolation-Theory

Published online by Cambridge University Press:  15 September 2014

E. T. Whittaker
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Extract

Let ƒ(x) be a given function of a variable x. We shall suppose that ƒ(x) is a one-valued analytic function, so that its Taylor's expansion in any part of the plane of the complex variable x can be derived from its Taylor's expansion in any other part of the plane by the process of analytic continuation.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 35 , 1915 , pp. 181 - 194
Copyright
Copyright © Royal Society of Edinburgh 1915

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References

page 188 note * It is not in general permissible to alter the order of the terms in a conditionally convergent series: but it may readily be proved that in the present case the value of the sum is not altered by the particular rearrangement which is made.

page 192 note * The manner in which the characteristic properties of the cardinal-function are required in order to ensure the vanishing of this remainder-term is very remarkable.

page 192 note † It should be noted that the interpolation-expansion considered is a “central-difference” formula, i.e. it makes use of all the tabulated values of f(x) both above and use is made only of the tabulated values of ƒ(x) for the values a, a + w, a + 2w, …. of the argument, and no use is made of the tabulated values of ƒ(x) for the values a − w, a − 2w, a − 3w, …. of the argument; in such cases a wholly different theorem holds, which I hope to give in a later paper.