How Do I Find the Bessel Transform of a Sequence of Numbers?

In summary, the conversation is about the proof of the recursion relations of Bessel functions and using differentiation to show the desired relation. The person asking for help has tried differentiating and replacing the value of nu, but is still unsure of what to do. The person offering help suggests using a specific differentiation method and mentions using it for a project on analog signal processing using Bessel functions and MATLAB.
  • #1
csmines
11
0
Hey guys I was wondering if you could help me out with a proof of the recursion relations of Bessel functions on my homework:

Show by direct differentiation that

[tex]


J_{\nu}(x)=\sum_{s=0}^{\infty} \frac{(-1)^{s}}{s!(s + \nu)!} \left (\frac{x}{2}\right)^{\nu+2s}

[/tex]

obeys the important recursion relations

[tex]
J_{\nu-1}(x)+J_{\nu+1}(x) = \frac{2\nu}{x}J_{\nu}(x)
[/tex]

[tex]
J_{\nu-1}(x)-J_{\nu+1}(x) = 2J_{\nu}(x)
[/tex]

I've tried differentiating with respect to x but I get a factor of 2s that's no good. And I've also tried replacing nu with nu plus one and nu minus one but that ends up with a lot of s terms as well. I am pretty much lost on what to do so if you could just point me in the right direction that'd be great. Thanks a lot.

Csmines
 
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  • #2
csmines,
I'm not at all sure, but the first step in this could be to show that
[tex]
\frac{d}{dx}\left[x^{-\nu}J_{\nu}(x)\right]=-x^{-\nu}J_{\nu+1}(x)
[/tex]
The trick is probably to
1) split off the s=0 term
2) make an index shift s->s+1 in the rest sum
3) differentiate (s=0 term vanishes).

This is IMO not too difficult. But can we use it to show the desired relation? I wonder.
 
Last edited:
  • #3
hey let me know how to find out bessel transform of a sequence of numbers ,as in we calculate Fourier transform of a sequence??

this is urgent pls do reply..
i need this for my project on "analog signal processing using bessel function using matlab"..
 

What are Bessel Functions?

Bessel functions are a special type of mathematical function named after the German mathematician Friedrich Bessel. They are widely used in physics and engineering to solve differential equations that arise in various applications, such as heat transfer, signal processing, and quantum mechanics.

What is the history of Bessel Functions?

Bessel functions were first introduced by Friedrich Bessel in 1816 while studying solutions to the differential equation governing the motion of a vibrating circular membrane. Since then, they have been extensively studied and applied in various fields of mathematics and science.

What are the properties of Bessel Functions?

Bessel functions have many important properties, including being solutions to certain differential equations, orthogonality relationships, and recurrence relations. They also have special values at certain points and can be expressed in terms of other mathematical functions.

What are the different types of Bessel Functions?

There are two main types of Bessel functions: the Bessel functions of the first kind (Jn) and the Bessel functions of the second kind (Yn). The Bessel functions of the first kind are used to represent solutions to problems with finite boundaries, while the Bessel functions of the second kind are used to represent solutions to problems with infinite boundaries.

What are the applications of Bessel Functions?

Bessel functions have a wide range of applications in various fields, including physics, engineering, and mathematics. They are used to model physical phenomena such as heat transfer, sound waves, and electromagnetic waves. They are also used in signal processing, image processing, and quantum mechanics.

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