Behavior of besselfunction for index

In summary: WolframAlpha examples. In summary, it can be said that for large x, the BesselJ(n,x) function behaves as an oscillating function with amplitude roughly = sqrt(2/(pi*x)), for n not very large, then the oscillations cease and the function decreases.
  • #1
Fabrice23
3
0
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.
untitled.jpg
 
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  • #2
Fabrice23 said:
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.
untitled.jpg

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 ?
 

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  • #3
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)
 
  • #4
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.
 
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  • #5
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.
 
  • #6
Fabrice23 said:
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 ?
 
  • #7
Fabrice23 said:
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)
 
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1. What is a Bessel function?

A Bessel function is a special type of mathematical function that is commonly used in physics and engineering. It is named after the German mathematician Friedrich Bessel and is defined as the solution to a certain differential equation.

2. What is the index of a Bessel function?

The index of a Bessel function is a parameter that determines the specific type of Bessel function being used. It is denoted by the letter ν (nu) and can be any real or complex number. The value of the index affects the shape and behavior of the Bessel function.

3. How does the index affect the behavior of a Bessel function?

The index of a Bessel function affects its behavior in several ways. The main effect is on the number and location of its zeros and extrema. A larger index will result in more zeros and extrema, and they will be spread out further along the x-axis. The index also affects the rate of decay of the Bessel function as x approaches infinity.

4. What are some applications of Bessel functions?

Bessel functions have a wide range of applications in physics and engineering. They are commonly used in solving problems involving wave phenomena, such as heat transfer, sound waves, and electromagnetic waves. They also arise in problems involving cylindrical or spherical symmetry, such as in fluid mechanics and quantum mechanics.

5. Can Bessel functions be used for non-integer indices?

Yes, Bessel functions can have non-integer indices. In fact, most applications involve non-integer indices. The solutions for integer indices are known as the Bessel functions of the first kind, while the solutions for non-integer indices are known as the Bessel functions of the second kind or the Neumann functions. Both types of Bessel functions have their own unique properties and applications.

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