Discussion Overview
The discussion revolves around the relationship between the Bessel functions of positive and negative orders, specifically how to compute Bessel[-v,x] given Bessel[v,x] in the context of numerical computations using Fortran. The scope includes theoretical aspects of Bessel functions and their properties.
Discussion Character
- Technical explanation
- Debate/contested
Main Points Raised
- One participant seeks assistance in finding the value of Bessel[-v,x] given Bessel[v,x], indicating a lack of understanding of their relationship.
- Another participant states that the Bessel function satisfies a specific differential equation and suggests that the sign of the order may not matter due to the squaring in the equation. They provide the relationship J_{-n}(x) = (-1)^nJ_n(x).
- A subsequent reply reiterates the relationship for integer orders but notes that complications arise when the order is not an integer.
- Further, it is mentioned that for non-integer orders, the relationship can be expressed as J_{-\nu} = cos(νπ)J_ν - sin(νπ)Y_ν, referencing Numerical Recipes.
- One participant acknowledges the cleverness of another's contribution without further elaboration.
Areas of Agreement / Disagreement
Participants agree on the relationship for integer orders but express uncertainty and difficulty regarding non-integer orders. The discussion remains unresolved regarding the general case.
Contextual Notes
The discussion highlights the complexity of the relationship between Bessel functions for non-integer orders and references specific mathematical expressions without resolving the implications of these relationships.