I What is the indefinite integral of Bessel function of 1 order (first k

[AhmedHesham]

Hi
When we find integrals of Bessel function we use recurrence relations.
But this requires that we have the variable X raised to some power and multiplied with the function .
But how about when we have Bessel function of first order and without multiplication?
How should we integrate it ?

 

[Orodruin]

It is easier if you write out explicitly in correct notation what you are asking about. It seems to me you are asking about what the primitive function of $J_n(x)$ is. In the general case, like with many other functions, there is no closed form solution to this.

 

[AhmedHesham]

It is easier if you write out explicitly in correct notation what you are asking about. It seems to me you are asking about what the primitive function of $J_n(x)$ is. In the general case, like with many other functions, there is no closed form solution to this.

The recurrence relations help writing the integrals of Bessel functions in terms of other Bessel functions of other orders.

 

[Orodruin]

The recurrence relations help writing the integrals of Bessel functions in terms of other Bessel functions of other orders.

Again, these are special cases.

 

[jasonRF]

But how about when we have Bessel function of first order and without multiplication?
How should we integrate it ?

Are you asking for $\int J_1(x) \, dx$? If so, then it has a particularly simple answer. Do you know a function $y$ such that $dy/dx = J_1(x)$ ? Hint: $y$ is a Bessel function.

Anyway, in general for complicated integrals with special function the first place I look is
https://dlmf.nist.gov/
the section on Bessel function includes a fair number of integrals.
Jason

 

[AhmedHesham]

Anyway, in general for complicated integrals with special function the first place I look is
https://dlmf.nist.gov/
the section on Bessel function includes a fair number of integrals.
Jason

Ok
Thanks
This really helps
I already solved it