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
>
>
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>
>
Andrzej Kozlowski
Toyama International University
JAPAN
http://sigma.tuins.ac.jp/
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