How to calculate this Bessel's terms limit?

In summary, Bessel's term limit is a predefined value of n=∞ in the formula for Bessel functions, which determines the maximum value of the index n. This limit is important in determining the convergence of Bessel functions and their use in various mathematical applications. It cannot be changed or modified, but is used in practical applications such as signal processing, image and sound analysis, and heat transfer. It is also significant in the study of differential equations, Fourier analysis, and wave phenomena in physics and engineering.
  • #1
JorgeM
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TL;DR Summary
I got to calculate this limit, but It is not clear to me if I have to use the original defitinion from the Bessels terms because I have already tried and It didn't resulted.
If you could suggest me a way to do it you would help me a lot.
Thanks
$$
\lim_{x \to 0} [ \frac{J_{p}(x)}{Y_{p}(x)} ]
$$
 
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  • #2
The problem I see is ##x=0## is a singular point of ##Y_p(x)## so the answer will depend on how one approaches ##x=0##. for ##p## an integer and approaching along the real ##+x## axis I think the limit is 0.
 
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1. How do I calculate the Bessel's terms limit?

To calculate the Bessel's terms limit, you can use the formula: lim n→∞ (Jn(x)/n) = 0, where Jn(x) is the Bessel function of the first kind and x is the variable. This formula is derived from the definition of the Bessel function and is used to determine the limit as the index n approaches infinity.

2. What is the significance of the Bessel's terms limit?

The Bessel's terms limit is important in the study of Bessel functions as it helps us understand the behavior of the function as the index n increases. It also allows us to approximate the Bessel function for large values of n, which can be useful in various applications such as in physics and engineering.

3. Can the Bessel's terms limit be calculated for negative values of n?

Yes, the Bessel's terms limit can be calculated for negative values of n. In this case, the formula becomes: lim n→-∞ (Jn(x)/n) = 0. This means that the limit is still equal to 0 as n approaches negative infinity, but the Bessel function used is now the modified Bessel function of the first kind, denoted as In(x).

4. Is there a specific method for calculating the Bessel's terms limit?

There is no specific method for calculating the Bessel's terms limit. It can be calculated using the formula mentioned in the first question, or by using properties of the Bessel function such as its recurrence relation. The method used may depend on the specific problem or application.

5. Can the Bessel's terms limit be used to find the limit of other functions?

No, the Bessel's terms limit can only be used to find the limit of Bessel functions. It cannot be applied to other types of functions as it is derived specifically for Bessel functions. However, other types of functions may have their own specific formulas or methods for finding their limit as the index approaches infinity.

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