Discussion Overview
The discussion centers on the indefinite integral of the Bessel function of the first order, specifically addressing the challenges of integrating this function without multiplication by a variable raised to a power. Participants explore the use of recurrence relations and the existence of closed-form solutions.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- Some participants note that integrating Bessel functions typically involves recurrence relations, which require multiplication by a variable raised to a power.
- Others suggest that there is no closed-form solution for the indefinite integral of the Bessel function of the first order, ##J_1(x)##, in the general case.
- One participant questions how to integrate ##J_1(x)## specifically, hinting that the derivative of a Bessel function might yield a simple answer.
- Another participant mentions that the Digital Library of Mathematical Functions (DLMF) contains a section on Bessel functions that includes various integrals, which could be useful for complicated integrals involving special functions.
Areas of Agreement / Disagreement
Participants express differing views on the existence of closed-form solutions for the integral of the Bessel function of the first order. While some acknowledge the utility of recurrence relations, others emphasize the lack of a general solution.
Contextual Notes
The discussion does not resolve the question of integrating the Bessel function without multiplication, and assumptions regarding the conditions under which recurrence relations apply remain unaddressed.