PiecewiseExpand and conditional results from Integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg78639] PiecewiseExpand and conditional results from Integrate
- From: dimitris <dimmechan at yahoo.com>
- Date: Thu, 5 Jul 2007 06:19:57 -0400 (EDT)
The following are based on a recent thread.
I found it good to collect the results.
(Version 5.2 is used)
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When Integrate returns conditional results (that is If[...]
structures)
it does not mean necessary that for the other values of the
parameter(s)
there is divergence or no closed form solution.
Below there is a nice example
In[10]:=
Integrate[E^(-x^2 - x*y)/Sqrt[x], {x, 0, Infinity}]
Out[10]=
If[Re[y] < 0, (E^(y^2/8)*Pi*Sqrt[-y]*(BesselI[-(1/4), y^2/8] +
BesselI[1/4, y^2/8]))/(2*Sqrt[2]),
Integrate[1/(E^(x*(x + y))*Sqrt[x]), {x, 0, Infinity}, Assumptions -
> Re[y] >= 0]]
At first someone might think that for Re[y]>0 there is no convergence
or closed form solution. However it is just that the integrator found
a result
which is valid for Re[y]<0. Nothing more, nothing less.
Specifying assumptions shows the situation
In[11]:=
(Integrate[E^(-x^2 - x*y)/Sqrt[x], {x, 0, Infinity}, Assumptions ->
#1[Re[y], 0]] & ) /@ {Greater, Less}
Out[11]=
{(1/2)*E^(y^2/8)*Sqrt[y]*BesselK[1/4, y^2/8], (E^(y^2/8)*Pi*Sqrt[-
y]*(BesselI[-(1/4), y^2/8] + BesselI[1/4, y^2/8]))/
(2*Sqrt[2])}
A nice application of PiecewiseExpand is given below
In[11]:=
Integrate[E^(-x^2 - x*y)/Sqrt[x], {x, 0, Infinity}]
PiecewiseExpand[%]
Out[11]=
If[Re[y] < 0, (E^(y^2/8)*Pi*Sqrt[-y]*(BesselI[-(1/4), y^2/8] +
BesselI[1/4, y^2/8]))/(2*Sqrt[2]),
Integrate[1/(E^(x*(x + y))*Sqrt[x]), {x, 0, Infinity}, Assumptions -
> Re[y] >= 0]]
Out[12]=
Piecewise[{{(E^(y^2/8)*Pi*Sqrt[-y]*(BesselI[-(1/4), y^2/8] +
BesselI[1/4, y^2/8]))/(2*Sqrt[2]), Re[y] <= 0}},
(1/2)*E^(y^2/8)*Sqrt[y]*BesselK[1/4, y^2/8]]
As another example consider
In[17]:=
Integrate[(1 + k*Sin[a]^2)^(1/2), {a, 0, 2*Pi}, Assumptions -> k
Reals]
PiecewiseExpand[%]
Out[17]=
If[k >= -1, 4*EllipticE[-k], Integrate[Sqrt[1 + k*Sin[a]^2], {a, 0,
2*Pi}, Assumptions -> k < -1]]
Out[18]=
4*EllipticE[-k]
which shows that although the If structure would indicate that there
is closed form
solution valid only for k>=-1, application of PiecewiseExpand shows
that the closed
form result is valid for all real k.
Dimitris