# Hamilton's Papers on Real Analysis

William Rowan Hamilton's most substantial paper on real analysis is On Fluctuating Functions, which is concerned largely with ideas from Fourier analysis. He wrote several short papers on topics in real analysis.

### `Fluctuating Functions'

On Fluctuating Functions (Transactions of the Royal Irish Academy, volume 19 (1843), pp. 264-321.)
In this paper, Hamilton sets out to explain the validity of the Fourier inversion formula by means of a principle which he calls the Principle of Fluctuation. He also uses this principle to obtain generalizations of the Fourier Inversion Formula. He also considers the representation of periodic functions by Fourier series, and discusses various applications of Fourier analysis.
On Fluctuating Functions [Abstract] (Proceedings of the Royal Irish Academy, 1 (1841), pp. 475-477.)
This is a short abstract of the above paper on Fluctuating Functions.

### Other Real Analysis Papers

On the Error of a received Principle of Analysis, respecting Functions which vanish with their Variables (Transactions of the Royal Irish Academy, volume 16, part 1 (1830), pp. 63-64.)
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.
Note to a Paper on the Error of a received Principle of Analysis (Transactions of the Royal Irish Academy, volume 16, part 2 (1831), pp. 129-130.)
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.
On Differences and Differentials of Functions of Zero (Transactions of the Royal Irish Academy, volume 17 (1837), pp. 235-236.)
In this paper, Hamilton generalizes a theorem of John Herschel, concerning the calculus of finite differences.
On certain discontinuous Integrals, connected with the Development of the Radical which represents the Reciprocal of the Distance between two Points (Philosophical Magazine, 20 (1842), pp. 288-294.)
In this paper, Hamilton considers the Taylor expansion of the reciprocal of the distance between two points, considered as a function of the distance of one of those points from the origin, obtaining expressions for the coefficients of this Taylor series, viewed as functions of the cosine of the angle between the two points as viewed from the origin.
On an Expression for the Numbers of Bernoulli, by means of a Definite Integral; and on some connected Processes of Summation and Integration (Philosophical Magazine, 23 (1843), pp. 360-367.)
In this paper, Hamilton obtains expressions for Bernoulli numbers involving definite integrals.
On the Calculation of the Numerical Values of a certain class of Multiple and Definite Integrals (Philosophical Magazine, 4th series, volume 14 (1857), pp. 375-382.)
This is the first installment of an projected series of papers on the numerical values of certain integrals. However no further installments were published.