What have "GenerateConditions -> False" done to a divergent integral?

Questions
,
1

Hello.

Copy and run the following 4-line code snippet and you’ll get the answers within 10 sec.

Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}]

Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity},GenerateConditions → False]

Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity},GenerateConditions → False] // N

Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, a, Infinity}] +4 Log[a] /. a → 10^-64 // N

The integral is divergent,seeing the first line of the program.

By adding “GenerateConditions → False”,seeing the second and the third line,we get a number which equals the result of the forth line.

My question is:what have “GenerateConditions → False” done to a divergent integral?

Thank you for your help.

2

There is a pole at x=0 (the integrand is infinity at 0)

    Limit[x^3*BesselK[2, x] BesselK[2, x], x -> 0]
    (*infinity*)

You can also see this by writing

Clear[a]
Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, a, Infinity}]

2226Untitled.png

Notice the Re[a]>0 in the above.

Now, why GenerateConditions -> False made it work? I do not know. I think this is a bug. But I am no expert on this. I do not think the value generated is even correct. Compare

Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}, GenerateConditions -> False] // N
(*.797059*)

to

    Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 10^(-64), Infinity}] // N
    (*590.259*)

Compare also what happens when using PrincipalValue -> True which tells it to ignore the simple pole at x=0

       Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}, PrincipalValue -> True]

it returns unevaluated. So I have no idea where this result

        Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}, GenerateConditions -> False]
        (*1/3 - 4 EulerGamma + Log[16]*)

came from.

3

With the setting GenerateConditions -> False, Integrate gives a regularized result (a result with a “singular part” removed).

4

Thanks for your answer.

Have you noticed that the result of the forth line just equals to 1/3 - 4 EulerGamma + Log[16] ?

How I find this?Using the following code.

Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, m, Infinity}]/ Log[m] /. m → 10^-16 // N

the answer is (-4.02163).(Of cousre,if you change 10^-16 to 10^-64,you’ll get -4.00541).

It shows that the integral and Log[m] are infinites of the same order when m=0.

I’m a rookie on this.I’m wondering that “GenerateConditions → False” can eliminate the divergent terms automatically.I do not know.

For further check.I have do the integral of x,x^2,x^-1,Log[x],and so on,which are obviously divergent. But “GenerateConditions → False” makes them equal to zero.

Integrate[x, {x, 0, Infinity},GenerateConditions → False]

Thanks again.

5

Haw-haw,that’s it! I really think so.

Thank you very much!

But can you tell me where I can find related informations.My MMA is 8.0 and the help document does not tell me things like this.Or has it been mentioned in the new vesions?

Thanks again.

6

With the setting GenerateConditions → False, Integrate gives a
regularized result (a result with a “singular part” removed).

Hello Daniel; In this case, can you please explain why then

   Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}, PrincipalValue -> True]

returned unevaluated? I thought the above is supposed to do pretty much what you said. In addition, I looked at help on GenerateConditions, and there is no mention of this side effect of its use. All what it says is that specifies that explicit conditions on parameters should be generated

7

(1) The implementation of PrincipalValuecatches many of the basic cases but is not intended as a general regularization.

(2) The meaning of GenerateConditions is very much overloaded in both the formal and connotational senses of that word. I doubt it is documented.