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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 1999.

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\centerline{\Largebf ON CONJUGATE FUNCTIONS,}

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\centerline{\Largebf OR ALGEBRAIC COUPLES}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (British Association Report, Edinburgh 1834,
   pp.\ 519--523.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 1999}

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\noindent
{\largeit On Conjugate Functions, or Algebraic Couples,
as tending to illustrate generally the Doctrine of Imaginary
Quantities, and as confirming the Results of Mr~Graves respecting
the Existence of Two independent Integers in the complete
expression of an Imaginary Logarithm.
By {\largesc W.~R. Hamilton}, M.R.I.A., Astronomer Royal for
Ireland.\par}

\bigskip

\centerline{[{\it Report of the Fourth Meeting of the
British Association for the Advancement of}}
\centerline{{\it Science; held at Edinburgh in 1834.}
(John Murray, London, 1835), pp.\ 519--523.]}

\bigbreak

Admitting, at first, the usual things about imaginaries, let
$$u + v \sqrt{-1} = \phi . (x + y \sqrt{-1}),
   \eqno {\rm (a.)}$$
in which $x$, $y$ are one pair of real quantities, and $u$, $v$
are another pair, depending on the former, and therefore capable
of being thus denoted, $u_{x,y}$, $v_{x,y}$.  It is easy to prove
that these two functions $u_{x,y}$, $v_{x,y}$, must satisfy the
two following equations between their partial differential
coefficients of the first order:
$${du \over dx} =   {dv \over dy},\quad
  {du \over dy} = - {dv \over dx}.
   \eqno {\rm (b.)}$$
Professor Hamilton calls these the {\it Equations of Conjugation},
between the functions $u$, $v$, because they are the necessary
and sufficient conditions in order that the imaginary expression
$u + v \sqrt{-1}$ should be a function of $x+ y \sqrt{-1}$.  And
he thinks that without any introduction of imaginary symbols, the
two real relations (b.), between two real functions, might have
been suggested by analogies of algebra, as constituting
between those two functions a connexion useful to study, and as
leading to the same results which are usually obtained by
imaginaries.  Dismissing, therefore, for the present, the
conception and language of imaginaries, Mr~Hamilton proposes to
consider a few properties of such {\it Conjugate Functions}, or
{\it Algebraic Couples\/}; defining two functions to be
{\it conjugate\/} when they satisfy the two equations of
conjugation, and calling, under the same circumstances, the pair
or couple $(u,v)$ {\it a function of the pair\/} $(x,y)$.

An easy extension of this view leads to the consideration of
relations between several pairs, and generally to reasonings and
operations upon pairs analogous to reasonings and operations on
single quantities.  For all such reasonings it is necessary to
establish definitions: the following definitions of sum and
product of pairs appear to Mr~Hamilton natural:
$$(x,y) + (a,b) = (x + a, y + b),
   \eqno {\rm (c.)}$$
$$(x,y) \times (a,b) = (xa- yb, xb + ya),
   \eqno {\rm (d.)}$$
and conduct to meanings of all integer powers and other rational
functions of pairs, enabling us to generalize any ordinary
algebraic equation from single quantities to pairs, and so to
interpret the research of all its roots, without introducing
imaginaries.

Without stopping to justify these definitions of sum and product
which will probably be admitted without difficulty, Mr~Hamilton
inquires what analogous meaning should be attached to an
exponential pair, or to the notation $(a,b)^{(x,y)}$; or,
finally, what form ought to be assigned to the conjugate
functions $u_{x,y}$, $v_{x,y}$, in the exponential equation
$$(a,b)^{(x,y)} = (u_{x,y}, v_{x,y}).
   \eqno {\rm (e.)}$$

In the theory of quantities, the most fundamental properties of
the exponential function $a^x = \phi(x)$ are these:
$$\phi(x) \phi(\xi) = \phi(x + \xi),
   \quad\hbox{and}\quad
  \phi(1) = a;
   \eqno {\rm (f.)}$$
Mr~Hamilton thinks it right, therefore, in the theory of pairs,
to establish by definition the analogous properties,
$$(a,b)^{(x,y)} (a,b)^{(\xi,\eta)} 
   =  (a,b)^{(x + \xi, y + \eta)},
   \eqno {\rm (g.)}$$
and
$$(a,b)^{(1,0)} = (a,b).
   \eqno {\rm (h.)}$$
Combining these properties with the equation (e.) and with the
definition (d.) of product, and defining an equation between pairs
to involve two equations between quantities, Mr~Hamilton obtains
the following pair of ordinary differential equations, or
equations in differences, to be combined with the two equations
of conjugation:
$$\left. \eqalign{
u_{x,y} v_{\xi, \eta} - v_{x,y} u_{\xi, \eta}
   &= u_{x + \xi, y + \eta},\cr
u_{x,y} v_{\xi, \eta} + v_{x,y} u_{\xi, \eta}
   &= v_{x + \xi, y + \eta},\cr}
   \right\}
   \eqno {\rm (i.)}$$
and also the following pair of conditions,
$$u_{1,0} = a,\quad v_{1,0} = b.
   \eqno {\rm (k.)}$$
Solving the pair of equations (i.), he finds
$$\left. \eqalign{
u_{x,y} &= f(\alpha' y + \beta' x) . \cos (\alpha y + \beta x),\cr
v_{x,y} &= f(\alpha' y + \beta' x) . \sin (\alpha y + \beta x),\cr}
   \right\}
   \eqno {\rm (l.)}$$
$\alpha$ $\beta$ $\alpha'$ $\beta'$ being any four constants,
independent of $x$ and $y$, and the function~$f$ being such that
$$f(r) = 1 + {r \over 1} + {r^2 \over 1.2} + {r^3 \over 1.2.3}
            + \hbox{\&c.};
   \eqno {\rm (m.)}$$
and having established the following, among many other general
properties of conjugate functions, that if two such functions be
put under the forms
$$\left. \eqalign{
u_{x,y} &= f(\rho_{x,y}) . \cos \theta_{x,y} \cr
v_{x,y} &= f(\rho_{x,y}) . \sin \theta_{x,y} \cr}
   \right\}
   \eqno {\rm (n.)}$$
$f$ still retaining its late meaning, the functions
$\rho_{x,y}$ $\theta_{x,y}$ are also conjugate, he concludes that
the 4 constants of (l.) are connected by these two relations
$$\beta'  = + \alpha,\quad
  \alpha' = - \beta,
   \eqno {\rm (o.)}$$
so that the general expressions for two conjugate exponential
functions are:
$$\left. \eqalign{
u_{x,y} &= f(\alpha x - \beta y) . \cos (\alpha y + \beta x),\cr
v_{x,y} &= f(\alpha x - \beta y) . \sin (\alpha y + \beta x);\cr}
   \right\}
   \eqno {\rm (p.)}$$
and it only remains to introduce the constants of the
{\it base-pair} $(a,b)$, by the conditions (k.).  These conditions
give
$$a = f(\alpha) . \cos \beta,\quad b = f(\alpha) . \sin \beta,
   \eqno {\rm (q.)}$$
and therefore, finally,
$$\left. \eqalign{
\alpha &= \int_1^{\sqrt{a^2 + b^2}} {dr \over r},\cr
\beta  &= \beta_0 + 2 i \pi,\cr}
   \right\}
   \eqno {\rm (r.)}$$
$i$ being an arbitrary integer, and $\beta_0$ being a quantity
which may be assumed as $> -\pi$, but not $> \pi$, and may then be
determined by the equations
$$\cos \beta_0 = {a \over \sqrt{a^2 + b^2}},\quad
  \sin \beta_0 = {b \over \sqrt{a^2 + b^2}}.
   \eqno {\rm (s.)}$$
The form of the direct exponential pair $(a,b)^{(x,y)}$, (or of
the direct conjugate exponential functions $u$, $v$,) is now
entirely determined; but the process has introduced {\it one\/}
arbitrary integer~$i$.

{\it Another arbitrary integer\/} is introduced by reversing the
process, and seeking the {\it inverse exponential\/} or
{\it logarithmic pair},
$$(x,y) = \mathop{\rm Log}\limits_{(a,b)} (u,v).
   \eqno {\rm (t.)}$$
Professor Hamilton finds for this inverse problem the formul{\ae}
$$x = {\alpha \rho + \beta \theta \over \alpha^2 + \beta^2},\quad
  x = {\alpha \theta - \beta \rho \over \alpha^2 + \beta^2};
   \eqno {\rm (u.)}$$
in which $\alpha$ $\beta$ are the constants deduced as before
by (r.) from the {\it base-pair\/} $(a,b)$, and involving the
integer $i$ in the expression of $\beta$; while $\rho$ and
$\theta$ are deduced from $u$ and $v$, with a new arbitrary
integer $k$ in $\theta$, by expressions analogous to (r.), namely,
$$\left. \eqalign{
\rho   &= \int_1^{\sqrt{u^2 + v^2}} {dr \over r},\cr
\theta &= \theta_0 + 2 k \pi,\cr}
   \right\}
   \eqno {\rm (v.)}$$
in which $\theta_0$ is supposed $> - \pi$ but not $> \pi$, and
$$\cos \theta_0 = {u \over \sqrt{u^2 + v^2}},\quad
  \sin \theta_0 = {v \over \sqrt{u^2 + v^2}}.
   \eqno {\rm (w.)}$$
By the definition of quotient, which the definition (d.) of
product suggests, the formul{\ae} (u.) may be briefly comprised in
the following expression of a logarithmic pair:
$$(x,y) = {(\rho, \theta) \over (\alpha, \beta)};
   \eqno {\rm (x.)}$$
and, reciprocally, the direct exponential pair $(u,v)$, as
already determined, may be concisely expressed by this other form
of the same equation,
$$(\rho, \theta) = (x,y)(\alpha, \beta),
   \eqno {\rm (y.)}$$
if we still suppose
$$\left. \eqalign{
(u,v) &= ( f(\rho)   . \cos \theta, f(\rho)   . \sin \theta ),\cr
(a,b) &= ( f(\alpha) . \cos \beta,  f(\alpha) . \sin \beta  ).\cr}
   \right\}
   \eqno {\rm (z.)}$$
Thus all the foregoing results respecting exponential and
logarithmic pairs may be comprised in the equations (y.) and (z.).

When translated into the language of imaginaries, they agree with
the results respecting imaginary exponential functions, direct
and inverse, which were published by Mr~Graves in the
{\it Philosophical Transactions\/} for 1829, and it was in
meditating on those results of Mr~Graves that Mr~Hamilton was
led, several years ago, to this theory of conjugate
functions,\footnote*{An Essay on this theory of Conjugate
Functions was presented some years ago by Professor Hamilton to
the Royal Irish Academy, and will be published in one of the next
forthcoming volumes of its {\it Transactions}.}
as tending to illustrate and confirm them.  For example,
Mr~Graves had found, for the logarithm of unity to the Napierian
base, the expression
$$\mathop{\rm Log}\limits_e . 1
   = { 2 k \pi \sqrt{-1} \over 1 + 2 i \pi \sqrt{-1} },$$
which is more general than the usual expression.  This result of
Mr. Graves appeared erroneous to the author of the excellent
Report on Algebra, which was lately printed for the Association;
but it is confirmed by Mr~Hamilton's theory, which conducts to it
under the form of a relation between real pairs, namely,
$$\mathop{\rm Log}\limits_{(e,0)} (1,0)
   = {(0, 2k\pi) \over (1, 2i\pi)}$$
and the connexion of this result with that Report was thought to
justify an greater fulness in the present
communication\footnote\dag{Since this communication was prepared,
Professor Hamilton has learned that Professor Ohm of Berlin has
been conducted by a different method to results respecting
Imaginary Logarithms, which agree with those of Mr~Graves: as do
also the results obtained in other ways, by Mons.~Vincent and by
Mr~Warren.  The partial differential equations (b.) have been
noticed and employed, for a different purpose, by Mr~Murphy of
Cambridge.}
that would have been proper otherwise on a question so abstract
and mathematical.

\bye