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MathGroup Archive 2007

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GenerateConditions setting

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74829] GenerateConditions setting
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Sat, 7 Apr 2007 04:01:42 -0400 (EDT)

Sorry fellas if I asked something which have been already answered,
but I
couldn't find anything relevant in the archives.

Anyway...

GenerateConditions: determines if conditions on the parameters should
be generated.
Default: Automatic
Admissible: True or False

Also from the Help Browser

(

FrontEndExecute[{HelpBrowserLookup["RefGuide", "GenerateConditions"]}]

)

we read

"The default setting is GenerateConditions->Automatic, which is
equivalent to a setting of True for one dimensional integrals".

So...

1)

In[3]:=
Integrate[Exp[a*x^2 + b*x^4], {x, -Infinity, Infinity},
GenerateConditions -> Automatic]
Out[3]=
If[Re[a] < 0 && Re[b] < 0, (Sqrt[-a]*BesselK[1/4, -(a^2/(8*b))])/
(E^(a^2/(8*b))*(2*Sqrt[-b])),
  Integrate[E^(a*x^2 + b*x^4), {x, -Infinity, Infinity}, Assumptions -
> Re[b] >= 0 || Re[a] >= 0]]

In[4]:=
Integrate[Exp[a*x^2 + b*x^4], {x, -Infinity, Infinity},
GenerateConditions -> True]
Out[4]=
If[Re[a] < 0 && Re[b] < 0, (Sqrt[-a]*BesselK[1/4, -(a^2/(8*b))])/
(E^(a^2/(8*b))*(2*Sqrt[-b])),
  Integrate[E^(a*x^2 + b*x^4), {x, -Infinity, Infinity}, Assumptions -
> Re[b] >= 0 || Re[a] >= 0]]

2)

In[5]:=
Integrate[x^=CE=B1*(Sin[x]/x - Cos[x])^2, {x, 0, Infinity},
GenerateConditions -> Automatic]
Out[5]=
If[Re[=CE=B1] < -1, 2^(-2 - =CE=B1)*((-4 + =CE=B1 - =CE=B1^2)*Gamma[-1 + =
=CE=B1] - 4*Gamma[=CE=B1,
0])*Sin[(Pi*=CE=B1)/2],
  Integrate[x^(-2 + =CE=B1)*((-x)*Cos[x] + Sin[x])^2, {x, 0, Infinity},
Assumptions -> Re[=CE=B1] >= -1]]

In[6]:=
Integrate[x^=CE=B1*(Sin[x]/x - Cos[x])^2, {x, 0, Infinity},
GenerateConditions -> True]
Out[6]=
If[Re[=CE=B1] < -1, 2^(-2 - =CE=B1)*((-4 + =CE=B1 - =CE=B1^2)*Gamma[-1 + =
=CE=B1] - 4*Gamma[=CE=B1,
0])*Sin[(Pi*=CE=B1)/2],
  Integrate[x^(-2 + =CE=B1)*((-x)*Cos[x] + Sin[x])^2, {x, 0, Infinity},
Assumptions -> Re[=CE=B1] >= -1]]

3)

In[7]:=
Integrate[Exp[(-s)*x]*x*BesselJ[0, x]*Cos[x], {x, 0, Infinity},
GenerateConditions -> Automatic]
Out[7]=
If[Re[s] > 0, (Sqrt[1 + 4/s^2]*s*Cos[(1/2)*ArcTan[2/s]] -
Sin[(3/2)*ArcTan[2/s]])/((1 + 4/s^2)^(3/4)*s^3),
  Integrate[(x*BesselJ[0, x]*Cos[x])/E^(s*x), {x, 0, Infinity},
Assumptions -> Re[s] <= 0]]

In[8]:=
Integrate[Exp[(-s)*x]*x*BesselJ[0, x]*Cos[x], {x, 0, Infinity},
GenerateConditions -> True]
Out[8]=
If[Re[s] > 0, (Sqrt[1 + 4/s^2]*s*Cos[(1/2)*ArcTan[2/s]] -
Sin[(3/2)*ArcTan[2/s]])/((1 + 4/s^2)^(3/4)*s^3),
  Integrate[(x*BesselJ[0, x]*Cos[x])/E^(s*x), {x, 0, Infinity},
Assumptions -> Re[s] <= 0]]

4) (this appeared recently)

In[11]:=
Integrate[x*BesselJ[0, x]*Cos[x], {x, 0, Infinity}, GenerateConditions
-> Automatic]
Out[11]=
0

In[12]:=
Integrate[x*BesselJ[0, x]*Cos[x], {x, 0, Infinity}, GenerateConditions
-> True]
Integrate::idiv: Integral of x*BesselJ[0, x]*Cos[x] does not converge
on {0,=E2=88=9E}
Out[12]=
Integrate[x*BesselJ[0, x]*Cos[x], {x, 0, Infinity}, GenerateConditions
-> True]

Consequently...

Contrary to what the help browser says the two setting are not
equivalent!
It's obvious that the setting GenerateConditions -> True does a more
extensive case.

Can someone provide me with any insight/explanation?

Thanks a lot.

Dimitris



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