From: fc03@ns1.cc.lehigh.edu (Frederick W. Chapman)
Newsgroups: sci.math
Subject: Re: help me with complex numbers ("double" and "dual" numbers)
Date: 21 Jan 93 15:43:32 GMT
Organization: Lehigh University

At last, a question dealing with *my* specialty -- Hypercomplex Function
Theory!  :-)

In article <1993Jan20.144533.19578@husc3.harvard.edu>,
mlevin@husc8.harvard.edu (Michael Levin) writes:

>     I am reading Kantor and Solodovnikov's "Hypercomplex Numbers"
>book, where he shows that there are three types of "complex" numbers.
>The first kind is the usual complex numbers where i^2 = -1.
>The second kind is the "double" numbers where i^2 = 1.
>The third kind is the "dual" numbers where i^2 = 0.

First, a small digression -- there are only three distinct ("non-
isomomorphic", as mathematicians say) *TWO* dimensional hypercomplex number
systems (usually called "finite-dimensional algebras" in more modern
terminology); however, there are many other such systems in *HIGHER*
dimensions.  In fact, if p(X) is a monic polynomial of degree n >= 2 with
real coefficients, we can form an n-dimensional hypercomplex system by
taking numbers of the form

                                2        3                n-1
             a  +  a  j  +  a  j  +  a  j  +  ...  +  a  j
              1     2        3        4                n

where we add componentwise and multiply using the distributive law,
reducing powers j^k for k>=n to linear combinations of 1, j, ..., j^{n-1}
via repeated applications of p(j) = 0.  For example, the classical complex
numbers correspond to p(X) = X^2 + 1, the double numbers to p(X) = X^2 - 1,
and the dual numbers to p(X) = X^2.  Such a hypercomplex system is
commutative, associative, and has a multiplicative identity.
(Mathematicians will recognize the system as "isomorphic" to the "quotient
algebra" R[X]/(p(X)).)


>I have found the definitions of functions like sines, cosines, logs,
>etc. in terms of the real and imaginary components, for the complex
>numbers in a book. I am now looking for a book which has these
>definitions for the other systems. So, does anyone know of a book
>which lists the definitions of the common functions in terms of the
>real and imaginary components for the double and dual numbers? I.e.,
>what is sin(z) when z=a+bi and i*i=0? That sort of thing. Any
>references will be appreciated. Please reply to
>mlevin@husc8.harvard.edu.
>
>Mike Levin
>
>(P.S. - I am not a math major; please take it easy).

You can define these functions using the usual power series definitions,
and then simplify the results using facts about the number systems.


DOUBLE NUMBERS:

The trick here is to make a change of basis.  Let epsilon_1 := (1 + i)/2
and epsilon_2 := (1 - i)/2; then {epsilon_1}^2 = epsilon_1, {epsilon_2}^2 =
epsilon_2, epsilon_1 * epsilon_2 = 0, and epsilon_1 + epsilon_2 = 1.  (We
say that epsilon_1 and epsilon_2 give a "resolution of the identity into
idempotents").  We can always write

               a + b i  =  (a+b) epsilon   +   (a-b) epsilon
                                        1                   2

Notice that the PRODUCT of two numbers of the form

                      c  epsilon   +  c  epsilon
                       1        1      2        2

and

                      d  epsilon   +  d  epsilon
                       1        1      2        2

is simply

                   c  d  epsilon   +  c  d  epsilon
                    1  1        1      2  2        2

Thus, the advantage of working with numbers expressed in terms of epsilon_1
and epsilon_2 is that we can add *AND* multiply COMPONENTWISE!  In the new
basis, we see that the double numbers are really just two copies of the
real numbers stuck together (what mathematicians call the "direct sum of
algebras").  When we apply a power series formula f(c) to c = c_1 epsilon_1
+ c_2 epsilon_2, we can apply the series to each component separately,
getting f(c) = f(c_1) epsilon_1 + f(c_2) epsilon_2; then, if you like, you
can make the reverse change of basis to get the result in terms of 1 and i.

Putting it all together, we get the following:

f(x+yi)  =  f({x+y}epsilon   +  {x-y}epsilon )
                          1                 2

         =  f(x+y) espilon   +  f(x-y) epsilon
                          1                   2

         =  (1/2){f(x+y) + f(x-y)}  +  (1/2){f(x+y) - f(x-y)} i

Fans of partial differential equations will notice that each component of
the result is a solution of the wave equation in one space- and one
time-dimension (i.e., satisfies (d/dx)^2 U - (d/dy)^2 U = 0).  One can in
fact develop an entire hypercomplex function theory over the algebra of the
double numbers in which the wave operator plays the same role that the
two-dimensional Laplacian plays in classical complex function theory.


DUAL NUMBERS:

The dual numbers are even easier to work with because the "generator" i has
a property mathematicians call "nilpotency"; a non-zero number is
"nilpotent" if some positive power of the number equals 0.  i^2 = 0 means
that i is "nilpotent of order 2".  This considerably simplifies the
binomial theorem:

            m     m       m-1                 m-2  2  2             m  m
   (x + y i)  =  x  +  m x    y i  +  m(m-1) x    y  i  +  ...  +  y  i
                                      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                                                    ZERO!

We can in turn use this to considerably simplify power series expressions
in z = x + y i, for example:

                                      2               3      
exp(x+yi) =  1 + (x+yi) + (1/2!)(x+yi)  + (1/3!)(x+yi)  + ... 

                           n
             + (1/n!)(x+yi)  + ...

                                  2                  3    2
          =  1 + (x+yi) + (1/2!)(x + 2xyi) + (1/3!)(x + 3x yi) + ...

                       n     n-1
             + (1/n!)(x  + nx   yi) + ...

                             2          3               n
          =  {1 + x + (1/2!)x  + (1/3!)x + ... + (1/n!)x + ...} +

                             2                    n-1
             {1 + x + (1/2!)x  + ... + (1/{n-1}!)x    + ...} yi

          =  exp(x) + exp(x) y i


In general, you will find that if f(z) is a power series expression in z =
x + y i, then f(z) = f(x) + (d/dx)f(x) y i.

One can also develop a function theory over the dual numbers, but it is
something of a degenerate case and not very interesting (to me).


I hope that this information will prove useful.

All the best to you,

Fred Chapman
Hypercomplex Function Theorist

-- 

o ------------------------------------------------------------------------- o
|  Frederick W. Chapman, User Services, Computing Center, Lehigh University |
|    Campus Phone:  8-3218     Preferred E-mail Address:  fc03@Lehigh.Edu   | 
o ------------------------------------------------------------------------- o
|             Ecstasy is transitory, but a theorem is forever!              |
o ------------------------------------------------------------------------- o
Newsgroups: sci.math