% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 1st June 1999.
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\centerline{\Largebf ON THE ERROR OF A RECEIVED PRINCIPLE}
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\centerline{\Largebf OF ANALYSIS, RESPECTING FUNCTIONS}
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\centerline{\Largebf WHICH VANISH WITH THEIR VARIABLES}
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\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
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\centerline{\largerm (Transactions of the Royal Irish Academy,
vol.~16, part~1, (1830), pp. 63--64.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 1999}
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{\largeit\noindent
On the Error of a received Principle of Analysis, respecting
Functions which vanish with their Variables.
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By {\largerm WILLIAM R. HAMILTON}, Royal Astronomer of Ireland,
\&c.}
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\centerline{Read January~25, 1830.}
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\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~16, part~1, (1830), pp. 63--64.]}
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It appears to be a received principle of analysis, that if a real
function of a positive variable ($x$) approaches to zero with the
variable, and vanishes along with it, then that function can be
developed in a real series of the form
$$A x^\alpha + B x^\beta + C x^\gamma + \hbox{\&c.},$$
the exponents $\alpha, \beta, \gamma,\ldots$ being constant and
positive, and the coefficients $A, B, C,\ldots$ being constant,
and all these constant exponents and coefficients being finite
and different from zero. This principle has been made the
foundation of important theories, and has not ever, so far as I
know, been questioned; but I believe that the following example
of exception, which would be easy to put in a more general form,
will sufficiently prove it to be erroneous; since if the
principle be true, it is by its nature universal.
The real function $e^{-x^{-2}}$, in which $e$ is the base of the
neperian logarithms, approaches to zero along with $x$ and
vanishes along with it. Yet if we could develope this function
in a series of the kind described, we should have
$$x^{-\alpha} e^{-x^{-2}}
= A + B x^{\beta - \alpha} + C X^{\gamma - \alpha} +
\hbox{\&c.}$$
in which we might suppose $\alpha$ the least exponent; and then,
while $x$ approached to $0$, the second member would tend to the
limit~$A$, which by hypothesis is different from $0$; and yet,
from the nature of exponential functions, the limit of the first
member is zero. We conclude, therefore, that the function
$e^{-x^{-2}}$ cannot be developed in a series of the kind
assumed, although it vanishes with its variable; and consequently
that, if we only know this property of a function, that it
vanishes when its variable vanishes, we cannot correctly assume
that it may be developed in such a series.
If any doubt should be felt respecting the truth of the remark,
that the function $x^{-\alpha} e^{-x^{-2}}$ tends to zero along
with $x$, when $\alpha$ is any positive constant, this doubt will
be removed by observing that the function $x^\alpha e^{x^{-2}}$,
which is the reciprocal of the former, increases without limit
while $x$ decreases to zero. For we may develop this latter
function, by the known theorems, in the essentially converging
series,
$$x^\alpha e^{x^{-2}}
= y^{-\alpha} e^{y^2}
= y^{-\alpha} + y^{2 - \alpha}
+ {y^{4 - \alpha} \over 2}
+ {y^{6 - \alpha} \over 2 \mathbin{.} 3}
+ {y^{8 - \alpha} \over 2 \mathbin{.} 3 \mathbin{.} 4}
+ \hbox{\&c.},$$
$y$ being the reciprocal of $x$; and while $y$ tends to
$+\infty$, the terms of this series remain all positive, and all
after a certain constant number increase indefinitely.
\bye