Re: More integration/ fullsimplify bugs in Mathematica 4.1
- To: mathgroup at smc.vnet.net
- Subject: [mg41722] Re: More integration/ fullsimplify bugs in Mathematica 4.1
- From: "Dr. Wolfgang Hintze" <weh at snafu.de>
- Date: Tue, 3 Jun 2003 07:13:04 -0400 (EDT)
- References: <bb4vt3$49a$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Mathematica 4.0 on Windows98: No bugs found in this example.
Interesting: Simplify gives simpler results than FullSimplify.
In[53]:=
ClearAll["Global`*"]
In[54]:=
f[x_] := Integrate[1/(a*x^2 + b*x + c/4)^n, x]
### Using (simple) Simplify
In[55]:=
Simplify[f[t]]
Out[55]=
1/(a*(-1 + n))*(2^(-1 - n)*(-b + Sqrt[b^2 - a*c] - 2*a*t)*
((b + Sqrt[b^2 - a*c] + 2*a*t)/Sqrt[b^2 - a*c])^n*
Hypergeometric2F1[1 - n, n, 2 - n,
(-b + Sqrt[b^2 - a*c] - 2*a*t)/(2*Sqrt[b^2 - a*c])])/
(c/4 + t*(b + a*t))^n
In[56]:=
Simplify[D[%, t]]
Out[56]=
(c/4 + t*(b + a*t))^(-n)
In[57]:=
Simplify[D[f[t], t]]
Out[57]=
(c/4 + t*(b + a*t))^(-n)
### Now, using FullSimplify
In[62]:=
FullSimplify[f[t]]
Out[62]=
1/a*(2^(-1 - n)*(b - Sqrt[b^2 - a*c] + 2*a*t)*
(1 + (b + 2*a*t)/Sqrt[b^2 - a*c])^n*Gamma[1 - n]*
Hypergeometric2F1Regularized[1 - n, n, 2 - n,
1/2 - (b + 2*a*t)/(2*Sqrt[b^2 - a*c])])/
(c/4 + t*(b + a*t))^n
In[63]:=
FullSimplify[D[%, t]]
Out[63]=
((-1 + n)*Gamma[1 - n]*
(-1 + (1/2 + (b + 2*a*t)/(2*Sqrt[b^2 - a*c]))^n*
(-1 + Gamma[2 - n])*Hypergeometric2F1Regularized[
1 - n, n, 2 - n, 1/2 - (b + 2*a*t)/
(2*Sqrt[b^2 - a*c])]))/(c/4 + t*(b + a*t))^n
### but still
In[65]:=
FullSimplify[D[f[t], t]]
Out[65]=
(c/4 + t*(b + a*t))^(-n)
Wolfgang
Richard Fateman wrote:
> Integrate[1/(a*x^2 + b*x + c/4)^n, x]
> FullSimplify[%]
> D[%,x]
> InputForm[FullSimplify[%]] ==>
>
> (4^n*(-1 + n)*Gamma[1 - n]*
> (-1 + (1/2 + (b + 2*a*x)/(2*Sqrt[b^2 - a*c]))^n*
> (-1 + Gamma[2 - n])*Hypergeometric2F1Regularized[
> 1 - n, n, 2 - n, 1/2 - (b + 2*a*x)/
> (2*Sqrt[b^2 - a*c])]))/(c + 4*True*(b + a*True))^n
>
> notice the denominator has some variable "True" in it.
>
> Cheers.
> RJF
>
>