Re: Boole does not work with symbolic limits in Integrate ?
- To: mathgroup at smc.vnet.net
- Subject: [mg73148] Re: Boole does not work with symbolic limits in Integrate ?
- From: "dimitris" <dimmechan at yahoo.com>
- Date: Sun, 4 Feb 2007 08:19:34 -0500 (EST)
- References: <eq1ihd$2ug$1@smc.vnet.net>
On Feb 3, 10:53 am, Mitch Murphy <m... at mix5.com> wrote:
> hey hey
>
> use of the boole indicator function inside integrate only works for
> numerical arguments, symbolic limits of integration just returns the
> unevaluated integral.
>
> simple example ...
>
> Integrate[2 u Boole[a < u], {u, 0, x}]
>
> out[] = Integrate[2 u Boole[a < u], {u, 0, x}]
>
> i would expect ...
>
> Integrate[2 u , {u, a, x}]
>
> out[] = x^2 - a^2
>
> but it does work with numerical limits of integration ...
>
> Integrate[2 u Boole[a < u], {u, 0, 2}]
>
> out[] = Piecewise[{{4, a <= 0}, {4 - a^2, 0 < a < 2 }]
>
> anybody know a trick to make this work ? i've tried a bunch of
> variations using evaluate[], logicalexpand[], piecewiseexpand[],
> assumptions->x Reals, ..., but nothing works.
>
> cheers,
> Mitch
Hello Martin.
Try
In[8]:=
Assuming[x > 0, Integrate[2*u*Boole[a < u], {u, 0, x}]]
Out[8]=
Piecewise[{{x^2, a <= 0 && x > 0}, {-a^2 + x^2, a > 0 && a - x < 0}}]
or
In[9]:=
Integrate[2*u*Boole[a < u], {u, 0, x}, Assumptions -> x > 0]
Out[9]=
Piecewise[{{x^2, a <= 0 && x > 0}, {-a^2 + x^2, a > 0 && a - x < 0}}]
or
In[10]:=
Block[{$Assumptions = x > 0}, Integrate[2*u*Boole[a < u], {u, 0, x}]]
Out[10]=
Piecewise[{{x^2, a <= 0 && x > 0}, {-a^2 + x^2, a > 0 && a - x < 0}}]
(*check*)
In[13]:=
Plot[Evaluate[% /. x -> 2], {a, -3, 3}]
(*plot to be displayed*)
In[14]:=
Integrate[2*u*Boole[a < u], {u, 0, 2}]
Out[14]=
Piecewise[{{4, a <= 0}, {4 - a^2, 0 < a < 2}}]
In[15]:=
Plot[Evaluate[%], {a, -3, 3}]
(*plot to be displayed*)
Best Regards
Dimitris