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MathGroup Archive 2000

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Re: Trying to define: Fractional Derivatives & Leibniz' display form for output and templates

  • To: mathgroup at smc.vnet.net
  • Subject: [mg22863] Re: Trying to define: Fractional Derivatives & Leibniz' display form for output and templates
  • From: "Kai G. Gauer" <gauer at sk.sympatico.ca>
  • Date: Sun, 2 Apr 2000 15:33:33 -0400 (EDT)
  • References: <8bhvta$noq@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Thanks for the replies, everybody, but unfortunately, right before I posted
this message, it seemed as though the Mathematica news server (at least on
my end) decided that well, it was just time to turn itself off for a week or
so. Anyways, if anyone's got an expiry ticket lingering on this post, send
me a mail and I'll repost. (I had been busy with tests and talks all week as
well, so I apologize to those of you who personally responded to my post...
I personally prefer replies of newsgroup posts to hit the newsgroup for all
others to see as well, but that's just me.....and yes, I also hate those
expiry posts because of server outages.... if only more people read the
archives/faq's...sigh...so different than answering the (pseudo)-daily
news).

As for the whole idea that I mentioned the exponents of style 1/2 in the
integral form of the gamma function, we can use binomial series expansions
to approximate (without worrying about the issue of convergence) the
function, and then its integral. The hard part is deciding when it converges
fast and when we should expect a risk of divergence.

And yes, I do know that integration and differentiation are essentially
"reverse" processes. Again, some power series expansions which converge (and
which converge absolutely or conditionally) do not necessarily always share
the same propeties under differentiation or integration. ie Try the harmonic
series partials:1/1+1/2+1/3+1/4+.....+1/x and taking its limit as x-> oo.
But then, what happens when we differentiate wrt to x first and then take
lim (x->oo), we get a conditional sum. Does anyone know how to test (or what
tests to use) to find when a series can be srtified to be abs., cond.
convergent, or divergent?

The reason I wonder about integration is because there are times then I like
to see how integral [f[y],dy, {x<y<b}] behaves as a function of x, where x
is an independent variable of the actual integration process, regardless of
how Mathematica decides to estimate indefinite integral of f(y).
Furthermore, I have no idea why WRI would define integration and
differentiation as completely different behaving operators. If they did
emphasize this, I would have to be surprised to say that they don't also
worry about issues such as why and when some functions can only be DEFINED
(rigourously, of course) on only pointwise continuous intervals or why there
are no built-in tests to say when the upper Riemann integral would actually
equate to the lower sum (I can construct examples for those of you who don't
know what I'm talking about).

Of course, I could also define a function which just decides to do a point
sample -> store to reference table -> plot for as possibly much as one
specific (and usually, a most uninteresting) interval. Unfortunately, this
is what actually would take time, and is actually where the execution
sequence of Mathematica code usually slows things down significantly.
However, as I said before, MOST functions that I usually work with in this
sense seem to behave "nice" (more or less) on many intervals, and that
point-by point sampling/plotting is usually not that bad an issue. What I
don't want, however, is a code which I have to rewrite every half our for
each zoom in of one particular interval to see if the curve really does try
to behave straight or regularly curvy, or whatever on most intervals.
There's gotta be more effective way to at least program an integration
engine process into some of the newer versions of Mathematica!

(and yes, thanks for the idea of compiling the integral as function-specific
lists, but it does remain to be function specific,and on a fcn for fcn
basis, I estimate that this will make it even SLOWER than the convergence
issue, since for EACH point, if we really were all that excited about how
bad the convergence actually was on the global selection of sample points
versus our few random choices, say 30 or so points to plot/curve, wouldn't
we also want to weight some test effort to see just how bad the randomness
factor could be when plotting the same function's derivative on two
different occasions but with different interval selections? This is why I
prefer choosing functions, as opposed to a group of randomly selected plot
points, as an aproximation to my operator's result. It's also why I
mentioned 1/2+1/3+1/6 as the example to sine instead of just telling it to
take the regular first derivative.)


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