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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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**bobbym****bumpkin**- From: Bumpkinland
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How can I represent

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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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**bobbym****bumpkin**- From: Bumpkinland
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May I see your M code and pardon my slow replies, I am having computer problems.

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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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**bobbym****bumpkin**- From: Bumpkinland
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What are you trying to do?

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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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**bobbym****bumpkin**- From: Bumpkinland
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The first integral seems to be close to 1 for |k| small and close to 0 for |k| large.

How did you determine that?

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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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**bobbym****bumpkin**- From: Bumpkinland
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What did you plot?

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

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**bobbym****bumpkin**- From: Bumpkinland
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Mathematica thinks

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

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**bobbym****bumpkin**- From: Bumpkinland
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No;

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

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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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**bobbym****bumpkin**- From: Bumpkinland
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How did you get that, mine just spits out a 1 for

See you later, I need to go offline.

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This works:

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

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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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**bobbym****bumpkin**- From: Bumpkinland
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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]}]]`

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

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**bobbym****bumpkin**- From: Bumpkinland
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What kind of answer are you expecting?

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