Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

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)*

Offline

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

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.**

Offline

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

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.**

Offline

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.

Offline

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

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.**

Offline

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

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.**

Offline

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

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.**

Offline

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

Offline

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

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.**

Offline

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

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.**

Offline

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.

Offline

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

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.**

Offline

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)*

Offline

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

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.**

Offline

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

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.**

Offline

Pages: **1**