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Limits of Bessel functions

Hi guys!

I'm currently writing a piece of work involving bessel functions J, K and I of order 0 and 1.

I need to find the values of K'(x)/K(x), I'(x)/I(x) ad J'(x)/J(x) around x=0. My supervisor suggested looking into methods like pinching and squeezing but really have no idea where to start.

Squeezing sounds a good idea but don't really know where to start, I just wondered if anyone could offer me any advice?
Reply 1
Original post by Crazy Horse
Hi guys!

I'm currently writing a piece of work involving bessel functions J, K and I of order 0 and 1.

I need to find the values of K'(x)/K(x), I'(x)/I(x) ad J'(x)/J(x) around x=0. My supervisor suggested looking into methods like pinching and squeezing but really have no idea where to start.

Squeezing sounds a good idea but don't really know where to start, I just wondered if anyone could offer me any advice?


Can't you just make use of their power series?
Reply 2
Original post by RichE
Can't you just make use of their power series?
A couple of these don't have a power expansion at x = 0 I think (and the limit isn't defined).
Reply 3
Original post by DFranklin
A couple of these don't have a power expansion at x = 0 I think (and the limit isn't defined).


No I'm very sure a power series wouldn't work. I did manage to find K'/K for order 0 using an upper bound and a lower bound I found in a research paper to 'squeeze' a for x->0 but it will only work for order 0 and I need something to work for order 1.

It's really annoying me because I'm an applied mathematician so analysis isn't my strong suit and isn't something I've studied in a long time but my supervisor seems to think it's something I should be able to crack out in like five minutes! :angry:
Reply 4
Crazy Horse
..
A power series works fine for at least half the cases (I think 4 of them but I can't be bothered to check again now). For the others, use the series expansion that involves a log term and take limits.

It really isn't a particularly involved problem; I've never studied Bessel functions and it only took me a few minutes to find suitable series expansions.
Reply 5
Original post by DFranklin
A power series works fine for at least half the cases (I think 4 of them but I can't be bothered to check again now). For the others, use the series expansion that involves a log term and take limits.

It really isn't a particularly involved problem; I've never studied Bessel functions and it only took me a few minutes to find suitable series expansions.


Are you talking about something along the lines of a Taylor expansion? Because that would involve the differentiation of K'/K surely?

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