Re: D vs. Derivative

• To: mathgroup at smc.vnet.net
• Subject: [mg15673] Re: D vs. Derivative
• From: "Allan Hayes" <hay at haystack.demon.co.uk>
• Date: Sat, 30 Jan 1999 04:29:05 -0500 (EST)
• References: <78pcjv\$d3r@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Gianluca Gorni wrote in message <78pcjv\$d3r at smc.vnet.net>...
>
>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.
>
>
> +---------------------------------+
> | 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 |
> +---------------------------------+
>
>

Gianluca,

Clear["`*"]

f' is  df& where df = D[f[#],#]

The problem that you mention seems to involve

h[x_] =HypergeometricPFQ[{I,1},{1+I},x];

D[h[#],#]

DirectedInfinity[((1/2 + I/2)*Sqrt[2]*
Sign[Gamma[2 + I]])/Sign[Gamma[1 + I]]]

which is equal to

h'[#]
DirectedInfinity[((1/2 + I/2)*Sqrt[2]*Sign[Gamma[2 + I]])/
Sign[Gamma[1 + I]]]

while

D[h[x],x]

-(1/((-1 + x)*x)*I*(1 -
HypergeometricPFQ[{I, 1}, {1 + I}, x] +
x*HypergeometricPFQ[{I, 1}, {1 + I}, x]))

Interestingly, from

feh[x_] = FunctionExpand[h[x]]

(I*Beta[x, I, 0])/x^I

We get

D[feh[#],#]

I/((1 - #1)*#1) + Beta[#1, I, 0]*#1^(-1 - I)

which is the same as

feh'[#]
I/((1 - #1)*#1) + Beta[#1, I, 0]*#1^(-1 - I)

and

D[feh[x],x]

I/((1 - x)*x) + x^(-1 - I)*Beta[x, I, 0]

Also

Sum[1/n^2,{n,Infinity}]

Pi^2/6

but

Sum[1/#^2,{#,Infinity}]

Sum[1/#1^2, {#1, 1, Infinity}]

Allan

---------------------
Allan Hayes
Mathematica Training and Consulting
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565

```

• Prev by Date: Re: How to construct all possible orderings
• Next by Date: Re: D vs. Derivative
• Previous by thread: Re: D vs. Derivative
• Next by thread: Re: D vs. Derivative