Hamilton published a short paper entitled
On the Error of a received Principle of Analysis, respecting Functions which vanish with their Variables.which appeared in the Transactions of the Royal Irish Academy, volume 16, part 1 (1830), pp. 63-64.
This paper is available in the following formats:
In this paper, Hamilton observes that the function equal to the exponential of -1/x2 for positive values of the real variable x cannot be expanded as a power series in x, despite the fact that the function tends to the limit zero as x tends to zero.
Hamilton subsequently published a note on this paper,
Note to a Paper on the Error of a received Principle of Analysis.which appeared in the Transactions of the Royal Irish Academy, volume 16, part 2 (1831), pp. 129-130.
This note is available in the following formats:
In this note Hamilton cites a paper by Cauchy, which had just come to his attention, in which the function he had considered, equal to the exponential of -1/x2 for positive values of the real variable x, was given as an example of a non-zero function with the property that the function and all its derivatives vanish at zero.
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D.R. Wilkins