On Thu, Jun 28, 2012 at 6:46 PM, Richard Fateman
<fateman at eecs.berkeley.edu>wrote:
> On 6/28/2012 12:00 PM, Stavros Macrakis wrote:
>
> There are two ways I can think of to introduce discontinuities into a
> function:
>
> * Functions with discontinuities, e.g. tan(x), 1/x, mod(x,2), etc. That
> reduces to the problem of solving for the appropriate values of the
> argument.
> * Explicit conditionals, e.g. if x>0 then 1 else 0.
>
> Everything else is a composition of these two.
>
> It is easy enough to recursively search the expression tree for these
> cases. There are some refinements to the basic idea that are harder to
> implement:
>
> * Dealing with removable or spurious discontinuities (if you want to),
> e.g. what is the desired result for sin(x)/x or mod(x,%pi)*sin(x)?
> * Combining discontinuities nicely, e.g. discontinuities(1/sin(x)) is
> %z1*%pi and discontinuities(1/sin(2*x)) is %z2*%pi/2; but the second case
> subsumes the first case, so discontinuities(1/(sin(x)*sin(2*x)) is simply
> %z3*%pi/2.
>
>
> discontinuities occur along branch cuts (which may be movable) e.g.
> consider log(x^2-1). Something
> bad happens at x=+1, x=-1, and along (some) line connecting them.
>
In the real case, log has a discontinuity at 0 (even though it's not
defined for arg <0). Even the purely real log(abs(x)) has a discontinuity
at 0.
I don't think the original poster was asking about the complex case, which
of course has other, um, complications.
The naive approach, spurious discontinuities and all, can be useful in many
cases.
-s
> If you have more than one variable, algorithms are lacking, generally.
>
> Mathematica has a "Reduce" command, sort of like Solve .... so that Reduce
> [ Log[x^2-1]-y==0, x]
>
> -\[Pi] < Im[y] <= \[Pi] && (x == -Sqrt[1 + E^y] || x == Sqrt[1 + E^y])
>
> which might be considered a description of the problematic locations for
> 1/(log(x^2-1)-y) ...
>
> Im(y) in [-pi,pi] and x=+- sqrt(1+exp(y))
>
> so y could be any value in some band it its complex plane, and
> x could be either of 2 points. I'm not sure why it restricts y, off hand.
>
>
>
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