Behavior of besselfunction for index

[Fabrice23]

How can I show that a first-degree Besselfunction a la Jn=Bessel(n,eta) for large etas always lead to something like a bathtub?
I.e. how can I show that J(n=eta) produces sth like a overshoot?
I would really appreciate every idea.

 

[JJacquelin]

How can I show that a first-degree Besselfunction a la Jn=Bessel(n,eta) for large etas always lead to something like a bathtub?
I.e. how can I show that J(n=eta) produces sth like a overshoot?
I would really appreciate every idea.

Sorry, I cannot understand your wording. Do you mean J(n=eta) = J(n,n) ?
If yes, where is no overshoot. The function BesselJ(n,n) is fairly regular.
Also, on your plot, what is drawn as a function of what ?

 

[Fabrice23]

Yes, I'mean J(n,n) n-->large (inf) (or take a look at http://literature.agilent.com/litweb/pdf/5954-9130.pdf pic. 23 d)

for large values the besselfunction starts to increase (Jn(n-->eta)) and than plunges rapidly but who can I explain that the values increase?
I plotted J(n,eta=2000) (and n ranged from -2100 to 2100)

 

[JJacquelin]

As I already said in my previous post, the function BesselJ(n,n) is without any overshot. It is a regular function continusely decreassing, without oscillation. See the graph in my previous post. This is confirmed by WolframAlpha :
http://www.wolframalpha.com/input/?i=plot+BesselJ(n,n)+from+n=1000+to+n=5000

Be careful in comparing BesselJ(n,n) and BesselJ(n, eta) if eta is constant and n variable. It is an oscillating function. Look at the example drawn by WolframAlpha :
http://www.wolframalpha.com/input/?i=plot+BesselJ(n,200)+from+n=0+to+n=300
In order to make it more clear, the value of eta is =100. We can see that for n>eta, the function is decreassing and tends to 0 when n tends to infinity. This is consistant with the asymptotic expansion of the BesselJ function.
Moreover, artifacts of ploting are likely to mistaken us. It is strange that your plot shows only positive values. Probably, roughly there are as many negative as positive points.

 

[Fabrice23]

JJacquelin: Actually my plot shows BesselJ(n, eta)^2 , What I don't get, why is the max value of BesselJ(n, eta)^2 rougly around eta?
Is there a way to describe the oscillating nature of the besselfunction because I'm interested in finding values that a 0.8*J0(eta)>Jn(eta)? (at the moment i use a MATLAB script but is there a way do discribe this with a formula that is easy to handle?)
Thanks for you help.

 

[JJacquelin]

Is there a way to describe the oscillating nature of the besselfunction because I'm interested in finding values that a 0.8*J0(eta)>Jn(eta)? (at the moment i use a MATLAB script but is there a way do discribe this with a formula that is easy to handle?).

Sorry I cannot understand your question :
"Is there a way to describe the oscillating nature of the Bessel function ?"
Suppose that the question was :
"Is there a way to describe the oscillating nature of the sine function?" What could be the answer ?
In both cases I think that the answer should be "It is an intinsic property of the function. The oscillating nature of the sine function (or the Bessel function) is described by the analytic definition of the function".
Certainly I misunderstand the meaning of your question.

What do you mean exactly by " finding values that a 0.8*J0(eta)>Jn(eta) " ? The values of what parameter or what wariable ?

 

[JJacquelin]

What I don't get, why is the max value of BesselJ(n, eta)^2 rougly around eta?

It is an intrinsic property of the Jn(x) function.
In case of constant and large x, the behaviour of Jn(x) as a function of n is known for a long time. Roughly, three domains are distinguished :

First domain: relatively small n (compared to large x).
Jn(x) is oscillating with amplitude roughly = sqrt(2/(pi*x))

Second domain: The transition.
As n increases, the amplitude of oscillation increases. For n not far from x, the oscillations cease. Then, with n increasing, Jn(x) begins to decrease.
In the transition domain, the asymptotic formulas are very complicated. A good compilation can be found in "Handbook of Mathematical Functions", M.Abramowitz, I.A.Stegun, Dover Publications, N.-Y., 1972, “Bessel functions of integer order", Ch.9, pp.355-389, especially “Asymptotic expansions in the transition region” pp.367-368.
It should be too complicated to deal with it on a forum. For even more explanation about those formulas, it should be necessary to consult some specialized documents listed p.368

Third domain : relatively large n (compared to x)
As n increases, Jn(x) decreases very rapidly. An asymptotic formula is :
Jn(x) roughly = (1/sqrt(2*pi*n))*(e*x/(2*n))^n
This formula is not valid on the transition domain. More accurate formulas can be found in the book referenced above.

The behaviour of the function Jn(n) is different (because representing for each n a particular point in the transition domain). This is a continuously decreasing function. An asymptotic formula is :
Jn(n) roughly=(1/Gamma(2/3))* (2/(9*n))^(1/3)

All above is for positive n.
For negative n, see the book referenced above (especially about oscillations in case of large negative n)