(simple) Bessel function integrals

zetafunc · 2017-01-21 08:55:57

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)

bobbym · 2017-01-21 11:03:48

How can I represent

?

zetafunc · 2017-01-21 11:07:33

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)

bobbym · 2017-01-21 11:22:02

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

zetafunc · 2017-01-21 11:32:53

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.

bobbym · 2017-01-21 11:38:39

What are you trying to do?

zetafunc · 2017-01-21 11:41:45

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)

bobbym · 2017-01-21 11:49:57

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

How did you determine that?

zetafunc · 2017-01-21 11:52:26

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.

bobbym · 2017-01-21 11:53:33

What did you plot?

zetafunc · 2017-01-21 12:00:31

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

bobbym · 2017-01-21 12:05:35

Mathematica thinks

zetafunc · 2017-01-21 12:07:19

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

bobbym · 2017-01-21 12:08:30

No;

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

zetafunc · 2017-01-21 12:32:04

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.

bobbym · 2017-01-21 12:36:42

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

See you later, I need to go offline.

zetafunc · 2017-01-21 22:01:35

This works:

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

zetafunc · 2017-01-21 22:05:06

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)

bobbym · 2017-01-22 00:14:40

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]}]]

zetafunc · 2017-01-22 00:18:56

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

bobbym · 2017-01-22 00:24:18

What kind of answer are you expecting?

Math Is Fun Forum

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

(simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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

Re: (simple) Bessel function integrals

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