Re: D vs. Derivative
- To: mathgroup at smc.vnet.net
- Subject: [mg15642] Re: [mg15601] D vs. Derivative
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Sat, 30 Jan 1999 04:28:39 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Hi Gianluca! (It's been a long time...) I noticed this sort of thing some time ago too. f' often seems to work better when you define your function using Function. In your case In[27]:= f =Function[x, Sum[ x^(n-1)/(n^3+n), {n, 1, Infinity} ]];f'[x] Out[27]= 1 1 -- (- (-1 + I + (1 - I) HypergeometricPFQ[{I, 1}, {1 + I}, 2 2 x x] + x HypergeometricPFQ[{1, 1 - I}, {2 - I}, x]) + Log[1 - x]) If you use patterns in your definition D[f[x],x] gives in fact a more complicated looking, although equivalent, answer,. Moreover, if you use the Function definition you D[f[x],x] also works and gives the same (complicated) answer as in the case of pattern based definiton. As I said above, I noticed some time ago that Function seems to work better and told my calculus students to define functions in this way rather than by using patterns. I would like to hear from someone from wri whether this is really justified or just based on a few "accidents". On Thu, Jan 28, 1999, Gianluca Gorni <gorni at dimi.uniud.it> wrote: >Hello! > >It seems that D[f[x],x] and f'[x] are not equivalent, and the latter can >give useless outputs. > >Consider the following power series, that converges in the unit disk of >the complex field: > > f[x_] = Sum[ x^(n-1)/(n^3+n), {n, 1, Infinity} ] > >With immediate assignment, f[x] is evaluated to a special function. >Suppose now that I need the derivative of f[x]. If I do it with > > D[ f[x], x ] > >there is no problem: I get a regular-looking special function >combination. But if I try to get the derivative with > > f'[x] > >the output is a formula containing DirectedInfinity. Moreover > > f'[x] // Simplify > >gives Indeterminate. > >By the way, the integral > > Integrate[ (1-Cos[y])/(E^y-x), {y, 0, Infinity} ] > >is left as it is by Mathematica, although it is equal to the special >function f[x] above, at least for many values of x. > >My version is 3.0.1 for PowerMacintosh. > > Gianluca Gorni > > > +---------------------------------+ > | Gianluca Gorni | > | Universita` di Udine | > | Dipartimento di Matematica | > | e Informatica | > | via delle Scienze 208 | > | I-33100 Udine UD | > | Italy | > +---------------------------------+ > | Ph.: (39) 0432-558422 | > | Fax: (39) 0432-558499 | > | mailto:gorni at dimi.uniud.it | > | http://www.dimi.uniud.it/~gorni | > +---------------------------------+ > > Andrzej Kozlowski Toyama International University JAPAN http://sigma.tuins.ac.jp/ http://eri2.tuins.ac.jp/