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#1 2017-01-21 08:55:57

zetafunc
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Registered: 2014-05-21
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(simple) Bessel function integrals

Can anyone suggest a closed form for either of these integrals?

and

where
denotes the Bessel function of the first kind,
is a positive integer,
is any real number, and
is constant.

There are various formulae out there for integrals of this kind: for instance, equation 5.52 in Gradshteyn-Ryzhik tells us that

and Wolfram's website (for instance, here) lists similar formulae, mostly involving generalised hypergeometric functions.

(I have been able to reduce the problems I have been posting in other threads and MathSE down to these integrals.)

Last edited by zetafunc (2017-01-21 09:10:03)

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#2 2017-01-21 11:03:48

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

How can I represent

?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2017-01-21 11:07:33

zetafunc
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Re: (simple) Bessel function integrals

You could just replace it with
I guess, for

The first integral seems to be close to 1 for |k| small and close to 0 for |k| large.

Last edited by zetafunc (2017-01-21 11:07:59)

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#4 2017-01-21 11:22:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

May I see your M code and pardon my slow replies, I am having computer problems.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2017-01-21 11:32:53

zetafunc
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Posts: 2,108
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Re: (simple) Bessel function integrals

If we leave out the
(which can easily be done via substitution), then this code:

Integrate[r^(-1) BesselJ[1, r], {r, t, Infinity}]

returns something involving Struve functions, which is not a closed form solution. However, those Struve functions can be bounded.

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#6 2017-01-21 11:38:39

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

What are you trying to do?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2017-01-21 11:41:45

zetafunc
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Posts: 2,108
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Re: (simple) Bessel function integrals

I would like to be able to either solve these integrals in terms of
explicitly, or bound these integrals by
for any

Last edited by zetafunc (2017-01-21 11:42:05)

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#8 2017-01-21 11:49:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

The first integral seems to be close to 1 for |k| small and close to 0 for |k| large.

How did you determine that?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2017-01-21 11:52:26

zetafunc
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Re: (simple) Bessel function integrals

For d = 2, I mean. I just tried it in Wolfram Alpha, and their plot of the integral seems to stay at 1 for almost all positive r.

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#10 2017-01-21 11:53:33

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

What did you plot?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2017-01-21 12:00:31

zetafunc
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Re: (simple) Bessel function integrals

Wolfram Alpha gives a plot of the integral here: http://www.wolframalpha.com/input/?i=in … %5B1,+x%5D

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#12 2017-01-21 12:05:35

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

Mathematica thinks


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2017-01-21 12:07:19

zetafunc
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Re: (simple) Bessel function integrals

I agree with that too -- did you use NIntegrate?

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#14 2017-01-21 12:08:30

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

No;

Integrate[BesselJ[1, x]/x, {x, 0, \[Infinity]}]

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2017-01-21 12:32:04

zetafunc
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Posts: 2,108
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Re: (simple) Bessel function integrals

Oh, I see. The definite integral returns:

ConditionalExpression[
 1/2 (2 + BesselJ[1, t] (2 - \[Pi] t StruveH[0, t]) + 
    t BesselJ[0, t] (-2 + \[Pi] StruveH[1, t])), 
 Re[t] > 0 && Im[t] == 0]

Here t = |k| so both conditional expressions are automatically satisfied. Unfortunately I can't see a nice way of bounding those Struve functions without getting a power of |k| that is too large.

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#16 2017-01-21 12:36:42

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

How did you get that, mine just spits out a 1 for

See you later, I need to go offline.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2017-01-21 22:01:35

zetafunc
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Registered: 2014-05-21
Posts: 2,108
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Re: (simple) Bessel function integrals

This works:

Integrate[BesselJ[1, x]/x, {x, t, \[Infinity]}]

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#18 2017-01-21 22:05:06

zetafunc
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Registered: 2014-05-21
Posts: 2,108
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Re: (simple) Bessel function integrals

Also

Integrate[BesselJ[d/2, x]/x, {x, t, \[Infinity]}]

returns

ConditionalExpression[
 2/d - 2^(-1 - d/2) t^(d/2)
    Gamma[d/
    4] HypergeometricPFQRegularized[{d/4}, {1 + d/4, 1 + d/2}, -(t^2/
     4)], Re[t] > 0 && Im[t] == 0]

Last edited by zetafunc (2017-01-21 22:05:21)

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#19 2017-01-22 00:14:40

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

Hi;

We can clean that up a bit if we make the assumption that t is real and greater than 0.

Assuming[t > 0, Integrate[BesselJ[d/2, x]/x, {x, t, \[Infinity]}]]

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#20 2017-01-22 00:18:56

zetafunc
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Registered: 2014-05-21
Posts: 2,108
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Re: (simple) Bessel function integrals

Hmm, that seems to return something multiplied by
. That is not good.

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#21 2017-01-22 00:24:18

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: (simple) Bessel function integrals

What kind of answer are you expecting?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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