# Asymptotic behaviour of bessel functions

• lavster
In summary, the speaker is learning about Bessel functions in their math course and is struggling with the concept. They mention a statement in their notes involving the Bessel function and an expression with a multiplying factor of exp. They also mention a mathematical trick or identity that may help explain this statement. They apologize for any confusion in their explanation.
lavster
Hi,

as part of my maths course i am learning about bessel functions. But this is something that I am not fully comfortable with - there seems to be a lot of tricks.

There is a statement in my notes that when $$\alpha_n>>1$$,
$$\frac{J_0(i^{\frac{3}{2}}\alpha_n\frac{r}{a})}{J_0(i^{\frac{3}{2}}\alpha_n)} = (\frac{r}{a})^\frac{1}{2}exp[-\sqrt{i}(1-\frac{r}{a}\alpha_n]$$

i know that $$\sqrt{i}=\pm\frac{1}{\sqrt{2}}(1+i)$$, which i think will help show this statement. I've also noticed that the argument in exp[] is the bit in the brackets of the bessl function in the denominator minus the bit in the brackets of the bessel function on the numerator. and the multiplying factor of exp in the RHS sems to me the bits in the brackets of the bessel function deivided by each other...

can anyone tell me if this is merely coincidence or whether there is a mathematical trick or identity that helps to show this?

sorry if this is confusing...

thanks, lav

there is meant to be an approximately equal sign between the first fraction (inbvolveing the bessel functions) and the fraction with r and a to the power of 1/2...

## 1. What are Bessel functions and what is their significance?

Bessel functions are a type of special functions that arise in many areas of mathematics and physics, particularly in problems involving cylindrical or spherical symmetry. They are named after the mathematician Friedrich Bessel and have applications in areas such as heat transfer, fluid mechanics, and wave propagation.

## 2. What is meant by asymptotic behavior of Bessel functions?

The asymptotic behavior of Bessel functions refers to the behavior of the functions as the argument (or input) approaches infinity. In other words, it describes how the functions behave at large values of their input. This behavior is important in understanding the properties and applications of Bessel functions.

## 3. How can Bessel functions be approximated for large values?

For large values of the argument, Bessel functions can be approximated using asymptotic expansions, which involve infinite series of simpler functions. These approximations are useful for making calculations and understanding the behavior of Bessel functions at large values.

## 4. What is the relationship between Bessel functions and other special functions?

Bessel functions are closely related to other special functions, such as the modified Bessel functions and the spherical Bessel functions. These functions have similar properties and are often used together in mathematical and physical equations.

## 5. What are some real-world applications of Bessel functions?

Bessel functions have a wide range of applications in physics, engineering, and other fields. Some examples include analyzing the vibration of a circular drumhead, describing the flow of heat in a cylindrical object, and understanding the behavior of electromagnetic waves in a cylindrical waveguide.

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