Bessel Function: Real v Parameter for y=0

[asdf1]

why does the v parameter in the bessel function
x^2y``+xy`+(x^2-v^2)y=0 have to be real and nonegative?

 

[Tide]

In the differential equation, there are no constraints on v (other than being finite) and solutions exist. If you want to call the solutions "standard Bessel functions" then the values of v are restricted. Generally, however, Bessel functions can have both complex arguments and indices.

 

[asdf1]

thank you very much!