Re: Integration Problems
- To: mathgroup at smc.vnet.net
- Subject: [mg13929] Re: Integration Problems
- From: "Allan Hayes" <hay at haystack.demon.cc.uk>
- Date: Tue, 8 Sep 1998 02:52:28 -0400
- References: <6svrm3$s40@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Georgios Kotsalis wrote in message <6svrm3$s40 at smc.vnet.net>...
>Dear Sirs ,
>
>while performing the following evaluation, i received an unexpected
>result:
>
>1.) W1[x_,t_]:=
> q[2,1][t] (Sin[4.73 x] - Cos[4.73 x] + Exp[-4.73 x] + Exp[-4.73
>(1-x)]);
>
>2.) W2[x_,t_]:=Evaluate[(D[W1[x,t],{x,2} ])^2]
>
>3.) Integrate[W2[x,t],{x,0,1}]
>
>Answer:
>
>(509.399 -2.922295 X 10^-5 I) q[2,1][t]^2
>
>Could you please tell me, why does an imaginary part arise though the
>function
>that is being integrated is real ?
>
Georgios:
It comes from internal inexact computations thet work with complex
forms. You can get rid of the small complex part by using Chop. I also
tried to integrate the exact form, with 4.73 replaced by 473/100, but
I had to turn off after quite a long time; however the answer came out
quite quickly when I expanded the integrand and integrated the summands
separately (this trick is often useful); also the inexact form treated
this way came out as real:
(1) Your example
W1[x_,t_]:=
q[2,1][t] (Sin[4.73 x] - Cos[4.73 x] + Exp[-4.73 x] + Exp[-4.73 (1-x)]);
W2[x_,t_]=D[W1[x,t],{x,2} ]^2;
Integrate[W2[x,t],{x,0,1}]
-15 2 (509.399 + 1.46147 10 I) q[2,
1][t]
Chop[%]
Out[4]=
2
509.399 q[2, 1][t]
(2) Expanded inexact form
Integrate[#,{x,0,1}]&/@Expand[W2[x,t]]
2
509.399 q[2, 1][t]
(3) Expanded exact form
Integrate[#,{x,0,1}]&/@Expand[Rationalize[W2[x,t]]]
2 2
105823817 q[2, 1][t] 39472283741 q[2, 1][t]
-(---------------------) + ----------------------- +
473/50 473/100
1000000 E 50000000 E
473 2 2
105823817 Cos[---] q[2, 1][t]
100
------------------------------- +
1000000
473 473 2
105823817 (Cos[---] - Sin[---]) q[2, 1][t]
100 100
------------------------------------------- -
1000000
473 473 2
105823817 (Cos[---] - Sin[---]) q[2, 1][t]
100 100
------------------------------------------- +
473/100
1000000 E
473 473 2
105823817 (Cos[---] + Sin[---]) q[2, 1][t]
100 100
------------------------------------------- +
1000000
473 473 2
105823817 (Cos[---] + Sin[---]) q[2, 1][t]
100 100
------------------------------------------- +
473/100
1000000 E
473 473 2
105823817 (--- - Sin[---]) q[2, 1][t]
50 50
-------------------------------------- +
4000000
473 473 2
105823817 (--- + Sin[---]) q[2, 1][t]
50 50
--------------------------------------
4000000
Simplify:
Simplify[%]
473/100 473/50 (105823817 (-100 + 746 E
+ 523 E +
473/50 473 473/50 473
200 E Cos[---] + 50 E Cos[---] +
100 50
473/100 473 2
200 E Sin[---]) q[2, 1][t] ) /
100
473/50
(100000000 E )
Numerical value
N[%]
2
509.399 q[2., 1.][t]
-------------------------------------------------------------
Allan Hayes
Training and Consulting
Leicester UK
http://www.haystack.demon.co.uk
hay at haystack.demon.co.uk
voice: +44 (0)116 271 4198
fax: +44(0)116 271 8642