Re: Integration problem
- To: mathgroup at smc.vnet.net
- Subject: [mg27452] Re: [mg27417] Integration problem
- From: BobHanlon at aol.com
- Date: Sun, 25 Feb 2001 20:55:59 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Tx1[i_] := ((1/a)*(1 - Abs[x - i*a]/a))*(UnitStep[x - (i - 1)*a] -
UnitStep[x - (i + 1)*a]);
Ty1[j_] := (UnitStep[y - (j - 1)*b] - UnitStep[y - j*b]);
You need to define L and W.
L = 5; W = 9;
JsxT = Table[Ty1[p]*Tx1[m], {m, L-1}, {p, W}];
JsxSum = Simplify[Sum[JsxT[[m, p]], {m, 1, 4}, {p, 1, 9}]];
The Simplify is used to speed up calculation of the following integral
Clear[soln];
soln[a_ /; a != 0, b_] :=
Evaluate[FullSimplify[Integrate[JsxSum, {x, 0, a}, {y, 0, b}],
Element[a, Reals]]] ;
Table[soln[a, b], {a, -3, 3}, {b, -3, 3}]
{{63/2, 21, 21/2, 0, 21/2, 21, 63/2}, {63/2, 21, 21/2, 0, 21/2, 21, 63/2},
{63/2, 21, 21/2, 0, 21/2, 21, 63/2},
{soln[0, -3], soln[0, -2], soln[0, -1], soln[0, 0], soln[0, 1], soln[0, 2],
soln[0, 3]}, {3/2, 1, 1/2, 0, 1/2, 1, 3/2},
{3/2, 1, 1/2, 0, 1/2, 1, 3/2}, {3/2, 1, 1/2, 0, 1/2, 1, 3/2}}
a /: a != 0 = True;
For a > 0,
Simplify[soln[a, b], a > 0] /. {UnitStep[a, b_] -> UnitStep[b],
UnitStep[-a, b_] -> 0} /. UnitStep[-b] -> UnitStep[b] - Sign[b]
(1/2)*b*Sign[b]
Since
FullSimplify[b*Sign[b] == Abs[b], Element[b, Reals]]
True
Then, soln[a_?Positive, b_] := Abs[b]/2;
For a <0,
FullSimplify[soln[a, b], a < 0] /. {UnitStep[a, b_] -> 0,
UnitStep[-a, b_] -> UnitStep[b]} /.
UnitStep[-b] -> UnitStep[b] - Sign[b]//Simplify
(21/2)*b*Sign[b]
Then,
Clear[soln];
soln[a_?Positive, b_] := Abs[b]/2;
soln[a_?Negative, b_] := 21*Abs[b]/2;
Bob Hanlon
In a message dated 2001/2/25 1:29:29 AM, drek1976 at yahoo.com writes:
>I am presently experiencing some problems with the use of integration in
>Mathematica, and I do not understand what could be wrong.
>I defined 2 functions as follow:
>
>Tx1[i_] := ((1/a)*(1 - Abs[x - i*a]/a))*(UnitStep[x - (i - 1)*a] -
>UnitStep[x - (i + 1)*a])
>Ty1[j_] := (UnitStep[y - (j - 1)*b] - UnitStep[y - j*b])
>
>where a, b are any assigned real values.
>
>I then formed a matrix as follows:
>
>JsxT := Table[Ty1[p]*Tx1[m], {m, L - 1}, {p, W}]
>
>which is then followed by:
>
>JsxSum = Sum[JsxT[[m,p]], {m, 1, 4}, {p, 1, 9}]
>
>I then try to integrate JsxSum with respect to x and y in the following
>way:
>
>Integrate[JsxSum, {x, 0, a}, {y, 0, b}]
>
>When Mathematica (I'm using version 4) tries to work out this last
>expression, it outputs the following message:
>
>"Unique::usym : -0.43661971830985913 is not a symbol or a valid sumbol
>name"
>
>before giving an output.
>
>This problem occurs whatever values I may use for a and b.
>Does someone know why?
>