# Simplifying a ratio of modified Bessel functions of the second kind

• Jul 15th 2011, 10:24 AM
penguinick
Simplifying a ratio of modified Bessel functions of the second kind
I found the following solution to a differential equation problem but need to simplify it:

K0(A*x)/K0(A*B)

where K0 is the modified Bessel function of the second kind with alpha = 0, x is the independent variable, and A and B are constants. I'm only interested in the interval x>=B.

So far, I've tried to use the definitions of the Bessel functions to replace K0s with I0s with J0s, or K0s with H0s with J0s, but I frequently wind up with zeroes top and bottom. Any tips or solutions?
• Jul 15th 2011, 11:46 PM
CaptainBlack
Re: Simplifying a ratio of modified Bessel functions of the second kind
Quote:

Originally Posted by penguinick
I found the following solution to a differential equation problem but need to simplify it:

K0(A*x)/K0(A*B)

where K0 is the modified Bessel function of the second kind with alpha = 0, x is the independent variable, and A and B are constants. I'm only interested in the interval x>=B.

So far, I've tried to use the definitions of the Bessel functions to replace K0s with I0s with J0s, or K0s with H0s with J0s, but I frequently wind up with zeroes top and bottom. Any tips or solutions?

If $\displaystyle A$ and $\displaystyle B$ are arbitrary constants in the solution, then set $\displaystyle C=1/K_0(A \times B)$ then you have $\displaystyle C \; K_0(Ax)$

CB
• Jul 16th 2011, 08:00 AM
penguinick
Re: Simplifying a ratio of modified Bessel functions of the second kind
Quote:

Originally Posted by CaptainBlack
If $\displaystyle A$ and $\displaystyle B$ are arbitrary constants in the solution, then set $\displaystyle C=1/K_0(A \times B)$ then you have $\displaystyle C \; K_0(Ax)$

CB

Nice... but there's no way to reduce it into something that doesn't have infinite series, I take it?