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Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp
#12
So we have to skip reals. At least formally it seems possible to define such function on hyperreals or superreals (second is more likely as they are discontinued by definition, with gaps) , but as I do not know enough to make such definition I will study a little from both Conway orginal book and Hyperreals, and Cantors ordinals/cardinals because these things does not work without them.

It seems that definition of functions on these numbers is not a very popular topic, most works try to link them to real functions as soon as possible, but I have seen brief mentioning of transcendental functions of surreals in the net, as well as surcomplex numbers so far made by simple adding of I to surreals.

Here is excerpt of one link I have started to read:

Nicolau C Saldanha on surreal functions

Quote:Unfortunately, I know of no written reference to this material.
I will do my best to reproduce here what I learned from Conway.

But first some warnings:

- There will be NO talk about continuity or differentiability of
surreal functions. In fact, this approach seems inadequate to
me, due to the existence of gaps in the ``surreal line''.

- Notice that series can NOT be added using epsilon-delta deffinitions
since that would make all non-trivial series diverge by falling
into a gap. For instance, we would not have 1/2 + 1/4 + 1/8 +... = 1,
since 1 - 1/omega is still larger than any partial sum.
When we write such things as 1 + omega^(-1) + omega^(-2) + ...
as the Cantor-Conway normal form of a surreal number, this ``series''
is NOT to be understood in the epsilon-delta sense, lest we get a
divergent series.

( By the way, 0.999... = 1 for the surreals ( since it holds for the
reals ) in the ``only'' meaningful sense the expressions have;
again, epsilon-delta definitions are out. )

- This does not use explicitly any form of summation like Cesaro's;
nor implicitly, as far as I can tell ( I could be wrong ).

- This DOES use the technique of defining things inductively, defining
x by means of x_L and x_R. This follows the same spirit as the
definitions of x + y, x * y, 1/x and the like.

And now a most crucial warning:

- The ( usual ) extentional notion of a function is NOT adequate here.
Two functions assuming the same values at each and every surreal number
must be considered DIFFERENT if the left and right options are not the
same. As an example, the functions:

f(x) = {|};

and

g(x) = { g(x_L) - 1 | g(x_R) + 1 };

are both constant equal zero. Usually, we would say that they are the
same function --- this is the extentional notion of what a function
is: a function is known if its extention, i.e., the values it assumes,
are known. Here, however, we have to adopt a radically different
point of view: in order to know a function, we have to know how it is
defined, and different definitions give different functions even if
the values coincide. In this sense, f and g are DIFFERENT.

An other way of looking at the situation, less radical but less
satisfactory, is to think of functions as having ``good'' and ``bad''
definitions. The above definition of f is probably ``good'', but
g is almost certanly ``bad''.

This is means we have to be careful about certain things we usually
take for granted. For instance, in a minute we will be defining log
as the integral of 1/x. In order to do this, the following definition
of 1/x is not satisfactory:

1/x = y iff xy = 1;

We now need a recursive, ``constructive'' definition of the form:

1/x = { ( stuff depending on x, x_L and x_R ) |
( other stuff ) };

such a definition is possible ( but not very easy ) to obtain.

Another danger: once we have log, we can't really invert it to get
exp: that would give us the values of exp, but not a definition we
could use later if we want to use exp to get new functions.

Now for the real action. We will adopt the convention:

f(x) = { f_L(x,x_L,x_R) | f_R(x,x_L,x_R) };

What follows is a definition of integration.

$
\int_a^b f(t) dt =

{
\int_a^{b_L} f(t) dt + \intd_{b_L}^b {f_L}(t) dt ,
\int_a^{b_R} f(t) dt + \intd_{b_R}^b {f_R}(t) dt ,
\int_{a_R}^b f(t) dt + \intd_a^{a_R} {f_L}(t) dt ,
\int_{a_L}^b f(t) dt + \intd_a^{a_L} {f_R}(t) dt
|
\int_a^{b_L} f(t) dt + \intd_{b_L}^b {f_R}(t) dt ,
\int_a^{b_R} f(t) dt + \intd_{b_R}^b {f_L}(t) dt ,
\int_{a_R}^b f(t) dt + \intd_a^{a_R} {f_R}(t) dt ,
\int_{a_L}^b f(t) dt + \intd_a^{a_L} {f_L}(t) dt
}
$

I used \TEX notation for integrals, subscripts and superscripts.
``\intd'' should be written as an integral sign with a capital `D'
over it, in the middle. It means direct integration, which means
do not chop the domain into pieces. Notice that some integrations
in the above definition will go from right to left, which means you
have to do the usual change of signs.

So now you know what log is! You define:

log(x) = \int_1^x 1/t dt;

This definition is good for any positive surreal number and satisfies
all the usual properties.

A similar definition can be found for the ``solution of the differential
equation''

dy/dt = g(t,y);

( Notice we are not *really* talking about derivatives )

from which definitions of exp , sin and cos can be obtained.

I will write down the definition later if someone wants me to, but I
think it is obvious if you understand the above definition of
integration.

I am not really sure of this, but I think gamma, bessel, zeta and
other functions can be done about as easily.

Exercise: Prove the Riemann Hypothesis for the surcomplex numbers.
( :-] )

Intuitively, however, it is relatively easy to see how to define
trigonometric functions for all surreals. Just say that TWOPI
is a period in the following surreal sense:

sin( x + TWOPI*n ) = sin(x);

for any surreal x and any n in the class Oz of surreal integers.

( Definition:

n is in Oz iff n = { n-1 | n+1 } )

The above definitions have this property. Notice that omega/TWOPI
is an integer, and therefore omega is a period.

Very roughly, the reason this works is the following: up to day omega
( exclusive ) you have been working only with finite numbers. When
the time comes to define cos(omega) and sin(omega) you still have NO
information about their values. What you do is of course pick the
simplest coherent answer ( this is what you do all the time with
surreal numbers ), and this is of course
cos(omega) = 1 and sin(omega) = 0.

By the same reasoning,

cos(omega^r) = 1,
sin(omega^r) = 0;

for any positive surreal number r.
( This is Cantor's exponentiation, not the analytic one )

By the way, there seems to be someone in Rutgers who is very interested
in this stuff. This someone may want to talk about this more directly.
Send e-mail or we might talk by telephone sometime.

--
Nicolau Corcao Saldanha


I have not any opinion about this, yet.


Ivars
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Messages In This Thread
RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/03/2008, 03:34 PM

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