Integral of Bessel J1 -> Struve?

[labaki]

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!

 

[JJacquelin]

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.

 

[labaki]

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?

 

[JJacquelin]

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.

 

[jasonRF]

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