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\centerline{\Largebf ON THE CALCULATION OF THE NUMERICAL}
\vskip12pt
\centerline{\Largebf VALUES OF A CERTAIN CLASS OF}
\vskip12pt
\centerline{\Largebf MULTIPLE AND DEFINITE INTEGRALS}
\vskip24pt
\centerline{\Largebf By}
\vskip24pt
\centerline{\Largebf William Rowan Hamilton}
\vskip24pt
\centerline{\largerm (Philosophical Magazine, 14 (1857), pp.\ 375--382.)}
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\vfill
\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 1999}
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\noindent
{\largeit On the Calculation of the Numerical Values of a certain
class of Multiple and Definite Integrals.
By\/} {\largerm Sir}
{\largesc William Rowan Hamilton}, {\largeit LL.D., M.R.I.A.,
F.R.A.S., \&c., Andrews' Professor of Astronomy in the
University of Dublin, and Royal Astronomer of
Ireland\/}\footnote*{Communicated by the Author.}.
\bigbreak
\vskip 12pt
\centerline{[{\it The London, Edinburgh and Dublin Philosophical
Magazine and Journal of Science,}}
\centerline{4th series, vol.~xiv (1857), pp.\ 375--382.]}
\bigskip
\centerline{\sc Section I.}
\nobreak\bigskip
[1.]
The results, in part numerical, of which a sketch is here to be
given, may serve to illustrate some points in the theory of
functions of large numbers, and in that of definite and multiple
integrals. In stating them, it will be convenient to employ a
notation which I have formerly published, and have often found to
be useful; namely the following,
$${\rm I}_t = \int_0^t dt;
\eqno (1)$$
or more fully,
$${\rm I}_t ft = \int_0^t ft \, dt;
\eqno (1)'$$
with which I am now disposed to combine this other symbol,
$${\rm J}_t = \int_t^\infty dt;
\eqno (2)$$
in such a manner as to write,
$${\rm J}_t ft = \int_t^\infty ft \, dt;
\eqno (2)'$$
and therefore
$${\rm I}_t + {\rm J}_t = \int_0^\infty dt.
\eqno (3)$$
I shall also retain, for the present, the known notation of
Vandermonde for factorials, which has been described and used by
Lacroix, and in which, for any positive whole value of $n$,
$$[x]^n = x (x - 1) (x - 2) \ldots (x - n + 1);
\eqno (4)$$
so that there are the transformations,
$$[x]^n = [x]^m [x - m]^{n-m} = [x]^{n+m} : [x - n]^m,
\enspace\hbox{\&c.};
\eqno (4)'$$
which are extended by definition to the case of null and negative
indices, and give, in particular,
$$[0]^{-n} = {1 \over [n]^n}
= {1 \over 1 \mathbin{.} 2 \mathbin{.} 3 \, \ldots \, n}.
\eqno (4)''$$
For example,
$$(1 + x)^n
= {\textstyle\sum\nolimits}_{m = 0}^{m = \infty}
[n]^m [0]^{-m} x^m.
\eqno (5)$$
It is easy, if it be desired, to translate these into other known
notations of factorials, but they may suffice on the present
occasion.
\bigbreak
[2.]
With the notations above described, it is evident that
$${\rm I}_t^n 1 = [0]^{-n} t^n;
\eqno (6)$$
and more generally, that
$${\rm I}_t^n t^m
= {t^{m+n} \over [m + n]^n} = [m]^{-n} t^{m+n}.
\eqno (6)'$$
Hence results the series,
$$(1 + {\rm I}_t + {\rm I}_t^2 + \cdots) 1
= (1 - {\rm I}_t)^{-1} 1 = e^t;
\eqno (7)$$
and accordingly, we have the finite relation,
$${\rm I}_t e^t = e^t - 1.
\eqno (7)'$$
The imaginary equation,
$$e^{t \sqrt{-1}} = (1 - \sqrt{-1} \, {\rm I}_t)^{-1} 1,
\eqno (8)$$
breaks up into the two real expressions,
$$\eqalignno{
\cos t &= (1 + {\rm I}_t^2)^{-1} 1, &(8)'\cr
\sin t &= {\rm I}_t (1 + {\rm I}_2^2)^{-1} 1. &(8)''\cr}$$
The series of Taylor may be concisely denoted by the formula,
$$f(x + t) = (1 - {\rm I}_t D_x)^{-1} fx;
\eqno (9)$$
and accordingly,
$${\rm I}_t D_x f(x + t) = {\rm I}_t f'(x + t) = f(x + t) - f(x).
\eqno (9)'$$
And other elementary applications of the symbol~${\rm I}_t$ may
easily be assigned, whereof some have been elsewhere indicated.
\bigbreak
[3.]
The following investigations relate chiefly to the function,
$$F_{n,r} t = {\rm I}_t^n (1 + 4 {\rm I}_t^2)^{-r-{1 \over 2}} 1;
\eqno (10)$$
or
$$F_{n,r} t = {\rm I}_t^n (1 + 4 {\rm I}_t^2)^{-r} ft,
\eqno (10)'$$
where
$$ft = F_{0,0} t = (1 + 4 {\rm I}_t^2)^{-{1 \over 2}} 1.
\eqno (11)$$
Developing by (5) and (6), and observing that
$$2^{2m} [- {\textstyle {1 \over 2}}]^m
= (-1)^m [2m]^m,
\eqno (12)$$
and that therefore
$$2^{2m} [- {\textstyle {1 \over 2}}]^m [0]^{-m} [0]^{-2m}
= (-1)^m ([0]^{-m})^2,
\eqno (12)'$$
we find the well-known series,
$$ft
= 1 - \left( {t \over 1} \right)^2
+ \left( {t^2 \over 1 \mathbin{.} 2} \right)^2
- \left( {t^3 \over 1 \mathbin{.} 2 \mathbin{.} 3} \right)^2
+ \hbox{\&c.},
\eqno (13)$$
which admits of being summed as follows,
$$ft = {2 \over \pi} \int_0^{\pi \over 2} d\omega \,
\cos (2t \cos \omega);
\eqno (13')$$
the function $ft$ being thus equal to a celebrated definite
integral, which is important in the mathematical theory of heat,
and has been treated by Fourier and by Poisson.
\bigbreak
[4.]
It was pointed out\footnote*{In his Second Memoir on the
Distribution of Heat in Solid Bodies, {\it Journal de l'Ecole
Polytechnique}, tome~xii.\ cahier~19, Paris,~1823, pages~349,
\&c.}
by the great analyst last named, that if there were written the
equation,
$$y = \int_0^\pi \cos (k \cos \omega) \, d\omega,
\eqno (14)$$
so that, in our recent notation,
$$y = \pi f \left( {k \over 2} \right),
\eqno (14)'$$
then for large, real, and positive values of $k$, the function
$y \surd k$ might be developed in a series of the form,
$$y \surd k
= \left( A + {A' \over k} + {A'' \over k^2} + \hbox{\&c.} \right)
\cos k
+ \left( B + {B' \over k} + {B'' \over k^2} + \hbox{\&c.} \right)
\sin k;
\eqno (15)$$
where a certain differential equation of the second order, which
$y \surd k$ was obliged to satisfy, was proved to be sufficient
for the successive deduction of as many of the other constant
coefficients, $A', A'',\ldots$ and $B', B'',\ldots$ of the
series, as might be desired, through an assigned system of
equations of condition, after the two first constants, $A$ and
$B$, were determined; and certain processes of definite
integration gave for them the following values,
$$A = B = \surd \pi;
\eqno (15)'$$
so that when $k$ is very large, we have nearly, as Poisson,
showed,
$$y \surd k = (\cos k + \sin k) \surd \pi.
\eqno (15)''$$
\bigbreak
[5.]
In my own paper on Fluctuating Functions\footnote*{In the
Transactions of the Royal Irish Academy, vol.~xix.\ part~2,
p.~313; Dublin 1843. Several copies of the paper alluded to were
distributed at Manchester in 1842, during the Meeting of the
British Association for that year: one was accepted by the great
Jacobi.},
I suggested a different process for arriving at this important
formula of approximation, (15)${}''$, which, with some slight
variation, may be briefly stated as follows. Introducing the two
definite integrals,
$$\left. \eqalign{
A_t &= {2 \over \pi} \int_0^{\pi \over 2} d\omega \,
\cos (2t \mathop{\rm vers} \omega),\cr
B_t &= {2 \over \pi} \int_0^{\pi \over 2} d\omega \,
\sin (2t \mathop{\rm vers} \omega),\cr}
\right\}
\eqno (16)$$
which give the following {\it rigorous transformation\/} of the
expression (13)${}'$, or of the function $ft$,
$$ft = A_t \cos 2t + B_t \sin 2t;
\eqno (16)'$$
and employing the {\it limiting values},
$$\left. \eqalign{
\lim\nolimits_{t=\infty} \mathbin{.} t^{1 \over 2} A_t
= {2 \over \pi} \int_0^\infty dx \, \cos (x^2)
&= (2\pi)^{-{1 \over 2}},\cr
\lim\nolimits_{t=\infty} \mathbin{.} t^{1 \over 2} B_t
= {2 \over \pi} \int_0^\infty dx \, \sin (x^2)
&= (2\pi)^{-{1 \over 2}};\cr}
\right\}
\eqno (16)''$$
(which two last and well-known integrals have indeed been used by
Poisson also,) I obtained (and, as I thought, more rapidly than
by his method) the following {\it approximate expression},
equivalent to that lately marked as (15)${}''$, for large, real,
and positive values of $t$:
$$ft = (\pi t)^{-{1 \over 2}} \sin \left( 2t + {\pi \over 4} \right);
\eqno (17)$$
which is sufficient to show that the {\it large and positive
roots\/} of the transcendental equation,
$$\int_0^{\pi \over 2} d \omega \, \cos (2t \cos \omega) = 0,
\eqno (17)'$$
are (as is known)\footnote*{It must, I think, be a misprint, by
which, in p.~353 of Poisson's memoir, the expression
$\displaystyle k = i\pi + {\pi \over 4}$,
is given, instead of
$\displaystyle k = i\pi - {\pi \over 4}$,
for the large roots of the transcendental equation $y = 0$. It
is remarkable, however, that this error of sign, if it be such,
does not appear to have had any influence on the correctness of
the physical conclusions of the memoir: which, no doubt, arises
from the circumstance that the number~$i$ is treated as infinite,
in the applications.}
very nearly of the form
$$t = {n\pi \over 2} - {\pi \over 8},
\eqno (17)''$$
where $n$ is a large whole number.
\bigbreak
[6.]
Poisson does not appear to have required, for the applications
which he wished to make, any more than the {\it two\/} constants,
which he called $A$ and $B$, of his descending series (15);
although (as has been said) he showed how all the {\it other\/}
constants of that series {\it could be successively\/} computed,
from them, if it had been thought necessary or desirable to do
so. In other words, he seems to have been content with assigning
the values (15)${}'$, and the formula (15)${}''$, as sufficient
for the purpose which he had in view. In my own paper, already
cited, I gave the {\it general term of the descending series\/}
for $ft$, by assigning a formula, which (with one or two
unimportant differences of notation) was the following:
$$(\pi t)^{1 \over 2} ft
= {\textstyle\sum\nolimits}_{m=0}^{m=\infty}
[0]^{-m} ([-{\textstyle {1 \over 2}}]^m)^2 (4t)^{-m}
\cos \left( 2t - {\pi \over 4} - {m \pi \over 2} \right).
\eqno (18)$$
As an example of the numerical approximation attainable hereby,
when $t$ was a moderately large number, (not necessarily whole,)
I assumed $t = 20$; and found that {\it sixty terms\/} of the
ultimately convergent, but initially divergent series (13), gave
$$\eqalignno{
f(20)
&= {2 \over \pi} \int_0^{\pi \over 2} d \omega \,
\cos (40 \cos \omega) \cr
&= + 7 \, 447 \, 387 \, 396 \, 709 \, 949 \cdot 965 \, 7957 \cr
&\mathrel{\phantom{=}}
- 7 \, 447 \, 387 \, 396 \, 709 \, 949 \cdot 958 \, 4289 \cr
&= + 0 \cdot 007 \, 3668;
&(19)\cr}$$
while only {\it three terms\/} of the ultimately divergent, but
initially convergent series (18) sufficed to give almost exactly
the same result, under the form,
$$\eqalignno{
f(20)
&= \left( 1 - {9\over 204800} \right)
{\cos 86^\circ \, 49' \, 52'' \over \sqrt{ 20\pi }}
+ {1 \over 320}
{\sin 86^\circ \, 49' \, 52'' \over \sqrt{ 20\pi }} \cr
&= 0 \cdot 0069736 + 0 \cdot 0003936
= + 0 \cdot 0073672.
&(19)'\cr}$$
\bigbreak
[7.]
The function $ft$ becomes infinitely small, when $t$ becomes
infinitely great, on account of the indefinite fluctuation which
$\cos (2t \cos \omega)$ then undergoes, under the sign of
integration in (13)${}'$; so that we may write
$$F_{0,0} \infty = f \infty = 0.
\eqno (20)$$
But it is by no means true that the value of this {\it other\/}
series,
$$F_{1,0} t = I_t ft
= {t \over 1}
- {t \over 3} \left( {t \over 1} \right)^2
+ {t \over 5} \left( {t \over 1 \mathbin{.} 2} \right)^2
+ \hbox{\&c.},
\eqno (21)$$
which may be expressed by the definite integral,
$$F_{1,0} t
= {1 \over \pi} \int_0^{\pi \over 2} d\omega \,
\sec \omega \, \sin (2t \cos \omega),
\eqno (21)'$$
is infinitesimal when $t$ is infinite. On the contrary, by
making
$$2t \cos \omega = x,\quad
d\omega \, \sec \omega
= - {dx \over x} \left( 1 - {x^2 \over 4t^2} \right)^{-{1 \over 2}},
\eqno (22)$$
the integral (21)${}'$ becomes, at the limit in question,
$$F_{1,0} \infty = {1 \over \pi} \int_0^\infty {dx \over x} \sin x
= {\textstyle {1 \over 2}}.
\eqno (21)''$$
Accordingly I verified, many years ago, that the series (21)
takes {\it nearly\/} this constant value, ${1 \over 2}$, when $t$
is a large and positive number. But I have lately been led to
inquire what is the {\it correction\/} to be applied to this
approximate value, in order to obtain a more accurate numerical
estimate of the function $F_{1,0} t$, or of the integral
$I_t ft$, when $t$ is large. In other words, having here, by (3)
and (21)${}''$, the {\it rigorous\/} relation,
$$F_{1,0} t = I_t ft = {\textstyle {1 \over 2}} - J_t ft,
\eqno (23)$$
I wished to evaluate, at least {\it approximately}, this
{\it other\/} definite integral, $- J_t ft$, for large and
positive values of $t$. And the result to which I have arrived
may be considered to be a very simple one; namely, that
$$- J_t ft = D_t^{-1} ft;
\eqno (24)$$
where $D_t^{-1} ft$ is a development analogous to the series
(18), and reproduces that series, when the operation $D_t$ is
performed.
\bigbreak
[8.]
As an example, it may be sufficient here to observe that if we
thus operate by $D_t$ on the function,
$$f^\backprime t
= \left( 1 - {129 \over 2^9 t^2} \right)
{\sin \left( 2t - {\pi \over 4} \right)
\over 2 \sqrt{\pi t}}
- {5 \cos \left( 2t - {\pi \over 4} \right)
\over 2^5 t \sqrt{\pi t}},
\eqno (25)$$
and suppress $t^{-{1 \over 2}}$ in the result, we are led to this
other function of $t$,
$$D_t f^\backprime t
= \left( 1 - {9 \over 2^9 t^2} \right)
{\cos \left( 2t - {\pi \over 4} \right)
\over \sqrt{\pi t}}
+ {\sin \left( 2t - {\pi \over 4} \right)
\over 2^4 t \sqrt{\pi t}};
\eqno (25)'$$
which coincides, so far as it has been developed, with the
expression (18) for $ft$: so that we may write, as at least
approxmimately true, the equation
$$f^\backprime t = D_t^{-1} ft.
\eqno (25)''$$
Substituting the value 20 for $t$, in order to obtain an
arithmetical comparison of results, we find
$$\eqalignno{f^\backprime (20)
&= \left( 1 - {129 \over 204800} \right)
{\sin 86^\circ \, 49' \, 52'' \over \sqrt{ 20\pi }}
- {\cos 86^\circ \, 49' \, 52'' \over 128 \sqrt{ 20\pi }} \cr
&= 0 \cdot 062942 - 0 \cdot 000054
= + 0 \cdot 062888;
&(26)\cr}$$
which ought, if the present theory be correct, to be nearly equal
to the definite integral, $- J_t ft$, for the case where $t =
20$. In other words, I am thus led to expect, after adding the
constant term ${1 \over 2}$, that the value of the connected
integral,
$$I_t ft
= \pi^{-1} \int_0^{\pi \over 2} d\omega \,
\sec \omega \, \sin (40 \cos \omega),
\eqno (26)'$$
must be nearly equal to the following number,
$$+ 0 \cdot 562888.
\eqno (26)''$$
And accordingly, when this last integral (26)${}'$ is developed
by means of the {\it ascending\/} series (21), I find that the
sum of the first sixty terms (beyond which it would be useless
for the present purpose to go) gives, as the difference of two
large but nearly equal numbers, (which are {\it themselves\/} of
interest, as representing certain {\it other\/} definite
integrals,) the value:
$$\eqalignno{
\pi^{-1} \int_0^{\pi \over 2} d\omega \,
\sec \omega \, \sin (40 \cos \omega),
&= + 3 \, 772 \, 428 \, 770 \, 679 \, 800 \cdot 537 \, 7058 \cr
&\mathrel{\phantom{=}}
- 3 \, 772 \, 428 \, 770 \, 679 \, 799 \cdot 974 \, 8177 \cr
&= + 0 \cdot 562 \, 888 \, 1;
&(26)'''\cr}$$
which can scarcely (as I estimate) be wrong in its last figure,
the calculation having been pushed to more decimals than are here
set down; and which exhibits as close an agreement as could be
desired with the result (26)${}''$ of an entirely different
method.
\bigbreak
[9.]
It must however be stated, that in extending the method thus
exemplified to higher orders of integrals, the development
denoted by $D_t^{-n} ft$, or the definite and multiple integral
$(-J_t)^n ft$, to which it is equivalent, comes to be {\it
corrected}, in passing to the {\it other\/} integral $I_t^n ft$,
not by a {\it constant term}, such as ${1 \over 2}$, but by a
{\it finite algebraical function}, which I shall here call
$f_n t$, and of which I happened to perceive the existence and
the law, while pursuing some unpublished researches respecting
vibration, a considerable time ago. Lest anything should prevent
me from soon submitting a continuation of the present little
paper, (for I wish to write on one or two other subjects,) let me
at least be permitted now to mention, that the spirit of the
process alluded to, for determining this finite and algebraical
{\it correction\/}\footnote*{Although this {\it algebraical
part}, $f_n t$, of the multiple integral $I_t^n ft$, is
{\it here\/} spoken of as a {\it correction\/} of the
{\it periodical part}, denoted above by $D_t^{-n} ft$, yet for
{\it large\/} and {\it positive values\/} of $t$ it is,
{\it arithmetically\/} speaking, by much the {\it most important
portion\/} of the whole: and accordingly I perceived (although I
did not publish) it long ago, whereas it is only very lately that
I have been led to {\it combine\/} with it the {\it
trigonometrical series}, deduced by a sort of extension of
Poisson's analysis.---When I thus venture to speak of any result
on this subject as being my own, it is with every deference to
the superior knowledge of other Correspondents of this Magazine,
who may be able to point out many anticipations of which I am not
yet informed. The formul{\ae} (27) (28) are perhaps those which
have the best chance of being new.},
$$I_t^n ft - (-J_t)^n ft = T_t^n ft - D_t^{-n} ft = f_n t,
\eqno (27)$$
(where $D_t^{-n} ft$ still denotes a descending and periodical
series, analogous to and including those above marked (18) and
(25),) consists in {\it developing the algebraical expression\/}
(10), (for the case $r = 0$, but with a corresponding development
for the more general case,) {\it according to descending powers
of the symbol\/}~$I_t$, and {\it retaining only those terms in
which the exponent of that symbol is positive or zero\/}: which
process gives the formula,
$$f_n t
= {\textstyle {1 \over 2}} I_t^{n-1}
(1 + 2^{-2} I_t^{-2})^{-{1 \over 2}} 1
= ( {\textstyle {1 \over 2}} I_t^{n-1}
- {\textstyle {1 \over 16}} I_t^{n-3}
+ {\textstyle {3 \over 256}} I_t^{n-5}
- \ldots ) 1;
\eqno (28)$$
that is, by (5) and (6),
$$f_n t
= {\textstyle\sum\nolimits}_{m=0}^{m=\infty}
2^{-2m-1} [- {\textstyle {1 \over 2}}]^m [0]^{-m}
[0]^{-(n - 2m - 1)} t^{n - 2m - 1};
\eqno (28)'$$
where the series may be written as if it were an infinite one,
but the terms involving negative powers of $t$ have each a null
coefficient, and are in this question to be suppressed.
For instance, I have arithmetically verified, at least for the
case $t = 10$, that the two finite algebraical functions,
$$\eqalignno{
f_6 t
&= {t^5 \over 240} - {t^3 \over 96} + {3t \over 256},
&(28)''\cr
f_7 t
&= {t^6 \over 1440} - {t^4 \over 384} + {3t^2 \over 512}
- {5 \over 2048},
&(28)'''\cr}$$
express the values of the two following sums or differences of
integrals,
$$\eqalignno{
f_6 t &= I_t^6 ft - J_t^6 ft, &(27)'\cr
f_7 t &= I_t^7 ft + J_t^7 ft; &(27)''\cr}$$
the calculations having been carried to several places of
decimals, and the integrals $I_t^6 ft$, $I_t^7 ft$ having each
been found as the difference of two large numbers.
\nobreak\bigskip
Observatory of Trinity College, Dublin,
September~29, 1857.
\nobreak\bigskip
\centerline{[To be continued.]}
\bye