On 12 August 2014 15:53, Waldek Hebisch <[email protected]> wrote: > Bill Page wrote: >> >> On 6 August 2014 21:13, Waldek Hebisch <[email protected]> wrote: >> > Bill Page wrote: >> >> I think that >> >> >> >> abs(x)/x ~= x/abs(x) >> >> >> >> If D(abs(x),x) = x/abs(x) then D(abs(x),[x,x]) = (abs(x)^2 - x^2 ) / >> >> abs(x)^3. >> >> This is zero only for x is real. >> > >> > But what D(abs(x),x) mean for non-real x? 'abs' considered >> > as function of complex variable is not differentiable. AFAICS >> > one reasonable interpretation of D(abs(x),x) is that it >> > automatically forces x to be real. >> >> Yes, D(abs(x),x) is not complex-differentiable. But D(abs(x),x) = >> x/abs(x) is nonetheless a derivative in the direction of the real >> axis. > > No, derivative in the direction of the real axis is real(x)/abs(x).
OK. > z/abs(z) = \partial_x abs(z) + i*\partial_y abs(z) where > z = x + iy. I believe that this is called a Wirtinger derivative (one of two related derivatives). Ref. http://en.wikipedia.org/wiki/Wirtinger_derivatives#Functions_of_one_complex_variable Strictly speaking I should write D(abs(z),x) = conjugate(z)/abs(z)/2 This remains suitable for my purpose. > Note that neither definition fits FriCAS. In > fact, once you try to use FriCAS to compute derivatives > of functions containig abs of complex argument you will > get nonsense regardless of what you take as D(abs(x), x). > More precisely, for a given function f you may be able to > tweak D(abs(x), x) so that D(f, x) will be sensible. > But for any value of D(abs(x), x) you will find some f > so that D(f, x) would be a nonsense. The reason is > that FriCAS assumes chain rule and Leibintz rule > and they are valid only for differentiable functions. I do not see any problem with Leibniz or chain rule provided we use both Wirtinger derivatves appropriately. http://en.wikipedia.org/wiki/Wirtinger_derivatives#Product_rule > >> My thinking is this: If we can "define" signum(x) as x/abs(x) >> then perhaps we can also "define" diracDelta(x) as (abs(x)^2 - x^2 ) / >> abs(x)^3 / 2 ? To do this the derivative D(abs(x),[x,x]) must not be >> zero. Of course these expressions are not actual distributions but >> perhaps we can also define integration of expressions involving abs so >> that we may use them to represent distributions in a consistent way. > > You probably can use (abs(x)^2 - x^2 ) / abs(x)^3 / 2 to > represent delta distribution, but there are plenty of alternative > representations, so I do not see why you want this one. > Also, connection of this with D(abs(x),[x,x]) is still > unclear. I would like derivative of expression to be compatible with derivative of distribution. In particular I noted the claim: "they can be readily extended to every space of generalized functions". > Concerning integration: if you want to integrate > something on _real_ interval, then integrator should > simplify your expression to zero since x is real on > te interval. ATM integrator is not doing this, but > sensible handling of integrals containing abs will > do this. So your representation is really unsuitable > for purpose of integrating delta distribution. > That is not clear to me. > Note that your expression probably would represent delta > via intagration along some complex contours. As contour I would choose the real line. > But we > do not want to use such contours in normal integration, > because results are discontinous and we would risk > wrong integrals for real function. Also, if delta > is represented explicitely, then integrating it > is quite easy. I do not remember formula by heart, > but it is relatively simple. > I would like to refer to http://axiom-wiki.newsynthesis.org/SandBoxDiracDelta Cheers, Bill Page. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/fricas-devel. For more options, visit https://groups.google.com/d/optout.
