Something about Bessel function

[xylai]

I am working on some numerical works. I use the computer language: Fortran language.
Here I have a problem about the Bessel functon.

Now I know the value of Bessel[v,x], where v is positive and real.
I want to know the value of Bessel[-v,x].

I don't know their relation. Can you help me?
Thanks!

 

[Born2bwire]

The Bessel function satisfies the differential equation,

$$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \nu^2)y = 0$$

We can see here that the sign of the order wold seem to be irrelevant because we take its square. However, the relationship is

$$J_{-n}(x) = (-1)^nJ_n(x)$$

 

[xylai]

The Bessel function satisfies the differential equation,

$$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \nu^2)y = 0$$

We can see here that the sign of the order wold seem to be irrelevant because we take its square. However, the relationship is

$$J_{-n}(x) = (-1)^nJ_n(x)$$

As far as I know, when n is integer, you are right: $$J_{-n}(x) = (-1)^nJ_n(x)$$.
But when n is not integer, it becomes very difficult.

 

[Born2bwire]

As far as I know, when n is integer, you are right: $$J_{-n}(x) = (-1)^nJ_n(x)$$.
But when n is not integer, it becomes very difficult.

$$J_{-\nu} =\cos (\nu\pi)J_\nu - \sin(\nu\pi)Y_\nu$$

via Numerical Recipes.

 

[xylai]

You are very clever.