The collected mathematical papers of Arthur Cayley.

568] NOTE ON THE INTEGRALS cos 2dx AND f sin x2 dx. 57
The error is in the assumption as to the limits of r, 0; viz. in the original
expressions for 4 (u2 - v2), 8uv, we integrate over the area of an indefinitely large square
(or rectangle); and the assumption is that we are at liberty, instead of this, to
integrate over the area of an indefinitely large circle.
Consider in general in the plane of xy, a closed curve, surrounding the origin,
depending on a parameter k, and such that each radius vector continually increases
and becomes indefinitely large as kc increases and becomes indefinitely large: the curve
in question may be referred to as the bounding curve; and the area inside or outside
this curve as the inside or outside area. And consider further an integral fzdxdy,
where z is a given function of x, y, and the integration extends over the inside area.
The function z may be such that, for a given form of the bounding curve, the integral,
as k becomes indefinitely large, continually approaches to a determinate limiting value
(this of course implies that z is indefinitely small for points at an indefinitely large
distance from the origin); and we may then say that the integral taken over the
infinite inside area has this determinate value; but it is by no means true that the
value is independent of the form of the bounding curve; or even that, being determinate for one form of this curve, it is determinate for another form of the curve.
I remark, however, that if z is always of the same sign (say always positive)
then the value, assumed to be determinate for a certain form of the bounding curve,
is independent of the form of this curve and remains therefore unaltered when we
pass to a different form of bounding curve. To fix the ideas, let the first form of
bounding curve be a square (x = +, y = + k), and the second form a circle (2 + y2 = 2).
Imagine a square inside a circle which is itself inside another square; then z being
always positive, the integral taken over the area of the circle is less than the integral
over the area of the larger square, greater than the integral over the area of the
smaller square. Let the sides of the two squares continually increase, then for each
square the integral has ultimately its limiting value; that is, for the area included
between the two squares the value is ultimately = 0, and consequently for the circle
the integral has ultimately the same value that it has for the square. When z is
not always of the same sign the proof is inapplicable; and although, for certain
forms of z, it may happen that the value of the integral is independent of the form
of the bounding curve, this is not in general the case.
We have thus a justification of the well known process for obtaining the value
of the integral | e-X dx, viz. calling this u, or writing
Jo
2u = e-2 dx,
then
4U2 = f e-(x2+y2) dx dy = e- rdrdO
7or. oo o 2
=27- r. 1, or u=1 4(Tr),

C. IX.

8