Re: integration of rational functions with parameter; simplification. Limit
- To: mathgroup at smc.vnet.net
- Subject: [mg41647] Re: integration of rational functions with parameter; simplification. Limit
- From: "Dr. Wolfgang Hintze" <weh at snafu.de>
- Date: Thu, 29 May 2003 08:14:24 -0400 (EDT)
- References: <bb1ujr$9ev$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Richard,
in Mathematica 4.0 I had none of the difficulties you mentioned.
Perhaps there is a slight difference in the definitions.
I started with the definition of the indefinite integral
g[t_]:= Integrate[(a*x+b)/(c*x^2+d*x+e)^n,x] /. x->t
Notice that I distiguished between the integration variable x and the
final variable t on which the result depends.
Now
g[t]
leads to the complicated expression you mentioned, and D[g[t],t] gives
back the expected integrand as a function of t.
Letting n=1; gives
g[t] =
d + 2 c t
(2 b c - a d) ArcTan[-----------------]
2
Sqrt[-d + 4 c e]
--------------------------------------- +
2
c Sqrt[-d + 4 c e]
2
a Log[e + d t + c t ]
---------------------
2 c
there are no error messages.
I don't know but perhaps the clue is to define the integral as I did
above, using a replacement in the end.
Regards,
Wolfgang
Richard Fateman wrote:
> in Mathematica 4.1, I tried integrating
>
> (a*x+b)/(c*x^2+d*x+e)^n
>
> and got a rather large answer involving AppelF1 and Hypergeometric
> functions. Indeed, differentiating that answer and
> FullSimplifying gets back to the start. So it seems to
> be An integral. There are, of course many such indefinite
> integrals, differing by a constant.
>
> So it is not impossible, but yet quite inconvenient, for
> the large answer to be useless (e.g. unbounded)
> for any particular value of n.
> It might even be useless for all values of n. Like
> substituting n=1 gives an error (1/0 generated)
> and even taking limits as n->1
> gives the peculiar result...
> (b*c*e*Infinity)/(Sqrt[Sign[d^2 - 4*c*e]]*
> Sign[-d + Sqrt[d^2 - 4*c*e]]*
> Sign[d + Sqrt[d^2 - 4*c*e]])
>
> This seems to suggest that if none of b,c,e are zero, the
> answer is Infinity; if one of them is zero, the answer is zero.
>
> If one tries FullSimplify[answer], and THEN takes
> the derivative, the path back to the original integrand
> seems to be blocked.
>
> What to do? It may be that the answer is bogus, having
> a "constant" that is somehow a singular function of n, or
> the simplification functions are inadequate (actually,
> not much question of this). The result of Limit could be
> better.
>
>