A way to handle the surcomplex exponential?
Here's an idea I had, I have no idea if this is standard or what. It's kind of trivial, but it seems worth mentioning at least.
EDIT: OK, here is somebody saying this previously; it may be standard, I'm not so sure.
So, background: Harry Gonshor defined the exponential function for the surreals. But what about for the surcomplex numbers? Say z=x+iy is surcomplex. It's clear what the magnitude of exp(z) should be (namely, exp(x)), but how is the direction determined? Now, if y is limited (i.e., not infinite), then cos(y) and sin(y) are defined and so one can define exp(z), and everything works out. But when y is infinite we have more of a problem; after you go around a circle infinitely many times, where do you end up? The oscillating nature of exp(iy) would seem to pose a problem for any attempt to sensibly define exp(iω).
So, here's my idea: Just chop off the infinite part of y. Define exp(iy)=exp(iy'), where y' is y with the infinite parts cut off.
And sure, it's a trivial thing to do, but it works pretty well. First off, we get the usual relation exp(z+w)=exp(z)exp(w); or at least we do assuming that sin and cos for limited surreals obey the usual relations, which they have to, right? (I don't have a text on surreals on hand at the moment.) Like I'm pretty sure that when you're proving exp(x+y)=exp(x)exp(y) for surreals, the hard part is if x and y may be infinite, not if they may be limited, because then you can use power series arguments adapted to the surreals (if I'm understanding/recalling correctly).
Secondly, let's look at the kernel of this map. For exponentiation on C, the kernel consists of numbers of the form τin (or 2πin, in the more usual notation), where n is an integer. Well, the commonly-used analogue in the surreals of the integers is the omnific integers -- those surreals with integral real part and no infinitesimal part. (That is to say, the infinite part has no effect on whether you're an omnific integer or not; all purely infinite surreals are omnific integers.) But that fits great here -- the kernel of this surcomplex exponentiation is precisely numbers of the form τin (or 2πin in the more usual notation) where n is an omnific integer. Really, we can just let all purely imaginary, purely infinite surcomplex numbers exponetiate to 1. Why not? Seems good to me!
And I mean seriously -- does anyone have a better idea? Well, maybe, but I haven't heard of it at least.
Does anyone know anything about this? Is this standard? Does this lead anywhere? It seems like it's too trivial to lead to anything seriously interesting beyond what having the surreal exponential (and sin and cos for limited surreals) already get you, but I mean just having a definition of the exponential for surcomplex numbers seems neat, so there you go.
-Harry
EDIT: OK, here is somebody saying this previously; it may be standard, I'm not so sure.
So, background: Harry Gonshor defined the exponential function for the surreals. But what about for the surcomplex numbers? Say z=x+iy is surcomplex. It's clear what the magnitude of exp(z) should be (namely, exp(x)), but how is the direction determined? Now, if y is limited (i.e., not infinite), then cos(y) and sin(y) are defined and so one can define exp(z), and everything works out. But when y is infinite we have more of a problem; after you go around a circle infinitely many times, where do you end up? The oscillating nature of exp(iy) would seem to pose a problem for any attempt to sensibly define exp(iω).
So, here's my idea: Just chop off the infinite part of y. Define exp(iy)=exp(iy'), where y' is y with the infinite parts cut off.
And sure, it's a trivial thing to do, but it works pretty well. First off, we get the usual relation exp(z+w)=exp(z)exp(w); or at least we do assuming that sin and cos for limited surreals obey the usual relations, which they have to, right? (I don't have a text on surreals on hand at the moment.) Like I'm pretty sure that when you're proving exp(x+y)=exp(x)exp(y) for surreals, the hard part is if x and y may be infinite, not if they may be limited, because then you can use power series arguments adapted to the surreals (if I'm understanding/recalling correctly).
Secondly, let's look at the kernel of this map. For exponentiation on C, the kernel consists of numbers of the form τin (or 2πin, in the more usual notation), where n is an integer. Well, the commonly-used analogue in the surreals of the integers is the omnific integers -- those surreals with integral real part and no infinitesimal part. (That is to say, the infinite part has no effect on whether you're an omnific integer or not; all purely infinite surreals are omnific integers.) But that fits great here -- the kernel of this surcomplex exponentiation is precisely numbers of the form τin (or 2πin in the more usual notation) where n is an omnific integer. Really, we can just let all purely imaginary, purely infinite surcomplex numbers exponetiate to 1. Why not? Seems good to me!
And I mean seriously -- does anyone have a better idea? Well, maybe, but I haven't heard of it at least.
Does anyone know anything about this? Is this standard? Does this lead anywhere? It seems like it's too trivial to lead to anything seriously interesting beyond what having the surreal exponential (and sin and cos for limited surreals) already get you, but I mean just having a definition of the exponential for surcomplex numbers seems neat, so there you go.
-Harry