Integral of Bessel J1 -> Struve?

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    Bessel Integral
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Discussion Overview

The discussion centers around the integral of the product of x and the Bessel function of the first kind of order 1, specifically the integral of x*J1(x) dx. Participants explore potential solutions, including the involvement of Struve functions, hypergeometric functions, and numerical methods. The context includes both definite and indefinite integrals.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents the integral of x*J1(x) dx and claims it results in a formula involving Struve functions.
  • Another participant suggests that solutions can also be expressed using hypergeometric functions or infinite series, noting that definite integrals might simplify the solution.
  • A later reply clarifies that simplification may not occur for arbitrary limits a and b, but could be possible for specific cases, such as a=0 and b=infinity.
  • Another participant shares their experience of needing numerical evaluation for the integral, indicating challenges in finding closed-form solutions.
  • There is a request for the professor's alternative solution, highlighting interest in differing definitions of "solution" and "simpler."

Areas of Agreement / Disagreement

Participants express differing views on the potential for simplification of the integral, with some suggesting that specific cases may allow for simplification while others maintain that no general simplification exists. The discussion remains unresolved regarding the existence of a simpler solution as proposed by the professor.

Contextual Notes

Limitations include the dependence on the specific values of a and b for definite integrals, as well as the unresolved nature of the professor's suggested simpler solution.

labaki
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Integral of Bessel J1 -> Struve?

Hello, everyone.

I have to solve the integral of x*J1(x) dx,
in which J1 is the Bessel function of first kind
and order 1. I found out that this results in

pi*x/2*[J1(x)*H0(x) - J0(x)*H1(x)],

in which H0 and H1 are Struve functions.
My prof told me Struve functions are not
necessary, though. He says I could get to a
much simpler solution, but didn't have time to
tell me which solution.

Do you guys have any idea?

Thanks!
 
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As far as I know, the ways to express the solutions are Struve functions, or hypergeometric functions, or infinite series. Nothing more.
Of course, if it was definite integral (for example 0 to infinity) the solution would be much simpler in some cases.
 


Hi, JJacquelin, thanks for your repply.
It is in fact a definite integral, say, from a to b.
Would there be any simplification in this case?
 


No simplification in case of any a and b.
Simplification might occur in some particular cases (for example a=0 and b=infinity). Each case requires a specific study (often difficult) in order to see if simplification is possible or not.
So, if the integral really is a definite integral, say, not with any a,b, but with well defined values, then show these values.
 


labaki said:
Hello, everyone.

I have to solve the integral of x*J1(x) dx,
in which J1 is the Bessel function of first kind
and order 1. I found out that this results in

pi*x/2*[J1(x)*H0(x) - J0(x)*H1(x)],

in which H0 and H1 are Struve functions.
My prof told me Struve functions are not
necessary, though. He says I could get to a
much simpler solution, but didn't have time to
tell me which solution.

Do you guys have any idea?

Thanks!

I have run into essentially the same integral in the past and have found nothing better. I ended up needed numerical evaluation of this anyway, so gave up on closed-form results eventually and resorted to more standard numerical techniques.

If your prof ever tells you his solution please share it with us - at the very least it will be interesting to see his definitions of "solution" and "simpler".

jason
 

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