gikiian said:My instructor told me that ##\nu## has to be a non-negative real number (i.e. ##\nu## ≥ 0). Do you have any idea why he said that?
Taking ν≥0 matters because it ensures the convergence of the solution to Bessel's ODE. This condition is known as the regularity condition and it guarantees that the solution is well-behaved and does not blow up at the origin. Without this condition, the solution may become unbounded and unusable in practical applications.
In Bessel's ODE, ν is a parameter that determines the type of solution. It is known as the order of the Bessel function and it affects the shape and behavior of the solution. For positive integer values of ν, the solution is a Bessel function of the first kind, while for non-integer values, it is a Bessel function of the second kind.
Taking ν≥0 restricts the values of ν to positive numbers, resulting in a specific type of solution known as the regular Bessel function. This function is well-behaved and has a finite value at the origin, making it useful in many applications such as heat transfer, vibration analysis, and electromagnetic theory.
Yes, the regularity condition can be relaxed for certain values of ν. For example, for negative integer values of ν, the solution is still well-behaved and finite at the origin. However, in most practical applications, it is necessary to take ν≥0 to ensure the convergence and usefulness of the solution.
The term ν squared in Bessel's ODE comes from the transformation of the ODE into a standard form for solving using power series or other analytical methods. This transformation simplifies the equation and makes it easier to find a solution. The squared term also arises from the properties of the Bessel function itself, which is a solution to many physical problems involving circular symmetry.