Question

(Bessel Functions) I'm trying to prove a property of Bessel Functions again, and I can't figure out what kind of algebra I should be doing to get to it. Could somebody throw me a hint without fully explaining it? (More info below.)

Answer 1

\[\text{Show that} \]\[(a) \ \ \ J'_1(x)=\frac{xJ_0(x)-J_1(x)}{x}\]

Answer 2

What are you allowed to start with? One way you could potentially do it is to just use the power series definition and manipulate it to give you what you want I think. I haven't done this before so I'm not sure but I've played with bessel functions a little in the past.

Answer 3

When you say "allowed", I dunno, I think you can do pretty much anything you want to it. So you're sure/confident that power series would do the job?

Answer 4

Well I mean it would most likely work but it might not be the prettiest method. Do you know any other properties of bessel functions that you've heard of recently that might help? I'm kinda vague on what their properties are other than they're a solution to sturm liouville problems almost like a damped sine and cosine wave

Answer 5

I know a ton of properties, I'm just not sure that we should be using them (there's absolutely no reason why not, but the first one I couldn't figure out, somebody solved independently of the common properties of bessel functions. Let me look them up, there's quite a few.

Answer 6

A very common/upfront one is that:

\[\Gamma(\alpha+1)=\alpha \Gamma(\alpha)\]

Answer 7

(Integers only, I think, normally nonnegative integers, and this is tied with the definition of factorial and is apparently part of the reason why 0!=1.

Answer 8

Have to check my profs' notes, not many properties are actually listed in the book at all.

http://i.imgur.com/CC2PU2n.png

http://i.imgur.com/2yMQULe.png

Answer 9

But yeah, it's the subtraction that's making me think twice about doing anything regarding shifting indices; What I tried and failed to make use of was Putting the whole set of terms on the RHS under a single summation index and factoring out (still inside the index) the common terms to see if any use could come of it, but I couldn't find any.

Answer 10

(Sorry, I just realized that I accidentally pasted properties of the Gamma Function, not the Bessel Function.)

Answer 11

That's ok :) I think those are going to be useful here anyway.
*Trying to work it out, grr*

Answer 12

Ooo I think I got something :O
Doing it the long power series way.

Answer 13

Sec, checking my work >.<

Answer 14

I did it backwards I guess.
From \(\Large\rm J_1'(x)=\dfrac{x J_0(x)-J_1(x)}{x}=J_0(x)-\frac{1}{x}J_1(x)\)

I wanted to first see what \(\Large\rm J_0(x)\) and \(\Large\rm \dfrac{1}{x}J_1(x)\) looked like.
Did some fiddling around to get the same power on x so I could combine the two series with subtraction and poof, \(\Large\rm J_1'(x)\) popped out.

Answer 15

Maybe I can upload a scan of my notes and you can look at them and see if the steps make sense.
Would be really difficult to type them out on here >.<

Answer 16

No worries, it's good to know someone has an idea. For the moment I'm working on another class, but-if you want-feel free to upload a scan of it. I'm just glad I at least know somebody figured it out.

Answer 17

Yah if you had uploaded Bessel Properties then maybe one/some of them lists a shorter way of doing this. But I think this works ok... if even a little tedious.