MHB Solutions to Bessel's equation with Wolfram

[Joppy]

Hi

Why does Wolfram find the solution of Bessels equation with non-integer $\nu$ to be of the form $y = c_1 J_\nu + c_2 Y_\nu$. I thought Bessels function of the second kind only arose in situations were $\nu$ is an integer. What's going on here? Thanks :).

As an example, see here.

 

[HallsofIvy]

I don't know where you got that idea. Bessel's equation, for any \nu, is a second order differential equation so has two independent solutions, one of which is singular at the origin. That one is the "Bessel's function of the second kind".

 

[Joppy]

I don't know where you got that idea.

I don't know where you got the idea that your response constitutes any sort of answer.

It stated in numerous texts that the general solution to Bessels equation with non-integer $\nu$ is given by, $y(x) = c_1 J_{\nu} + c_2 J_{-\nu}$, two linearly independent solutions.

Bessel's equation, for any \nu, is a second order differential equation so has two independent solutions, one of which is singular at the origin. That one is the "Bessel's function of the second kind".

When Bessels function of the second kind is introduced we then say that a general solution for all x>0 is $y(x) = c_1 J_{\nu} + c_2 Y_{\nu}$, as you have said. I realize that $Y_{\nu}$ includes both Bessels function of the first kind of order $\nu$ and $\-nu$, but this doesn't make them equivalent.

So, again, my question. Why, in the link in the OP, is the solution not of the form $y(x) = c_1 J_{\nu} + c_2 J_{-\nu}$, since $\nu$ is non-integer.