Exploring the Relationship Between g(x,t) and J_n(x)

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In summary, the conversation discusses using Bessel functions to show a series of terms and how to show that 1 can be represented as a sum of squares of Bessel functions. The concept of Bessel functions being orthogonal is mentioned as a potential approach to solving the problem.
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\(\displaystyle g(x,t) = e^{(\frac{x}{2})(t-\frac{1}{t})}=\sum_{n=-\infty}^{\infty}J_{n}(x)t^{n}\)

and

\(\displaystyle \left| J_{0}(x) \right|\le 1 \) and \(\displaystyle \left| J_{n}(x) \right|\le \frac{1}{\sqrt{2}} \)

how to show that

1=\(\displaystyle (J_{0}(x))^{2}+2(J_{1}(x))^{2}+2(J_{2}(x))^{2}+...\)

I don't have idea
 
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Re: [please help]Bessel function

Another said:
\(\displaystyle g(x,t) = e^{(\frac{x}{2})(t-\frac{1}{t})}=\sum_{n=-\infty}^{\infty}J_{n}(x)t^{n}\)

and

\(\displaystyle \left| J_{0}(x) \right|\le 1 \) and \(\displaystyle \left| J_{n}(x) \right|\le \frac{1}{\sqrt{2}} \)

how to show that

1=\(\displaystyle (J_{0}(x))^{2}+2(J_{1}(x))^{2}+2(J_{2}(x))^{2}+...\)

I don't have idea
I haven't done the whole thing but note that Bessel functions are orthogonal:
\(\displaystyle \int _0 ^1 J_{\nu} \left ( \alpha_{\nu ~ m} x \right ) ~ J_{\nu } \left ( \alpha_{\nu ~ n} x \right ) ~x ~dx = \frac{1}{2} \left [ J_{\nu + 1} \left ( \alpha _{\nu ~ m} \right ) \right ] ^2 \delta_ {m~n}\)

Does this give you any ideas?

-Dan
 

FAQ: Exploring the Relationship Between g(x,t) and J_n(x)

1. What is the definition of g(x,t)?

g(x,t) is a mathematical function that represents the relationship between two variables, x and t. It is often used in physics and engineering to describe the behavior of a system over time.

2. How is g(x,t) related to Jn(x)?

The function Jn(x) is a special type of mathematical function known as a Bessel function. It is used to solve differential equations that arise in problems involving waves and vibrations. The relationship between g(x,t) and Jn(x) is that g(x,t) can be expressed as a linear combination of Jn(x) functions.

3. What is the significance of the variable n in Jn(x)?

The variable n in Jn(x) represents the order of the Bessel function. It determines the number of nodes (points of zero amplitude) in the function and affects its overall shape. Different values of n will result in different solutions to the same problem.

4. How is g(x,t) used in real-world applications?

g(x,t) has many applications in physics and engineering, particularly in the study of waves and vibrations. It can be used to describe the behavior of sound waves, electromagnetic waves, and mechanical vibrations. It is also used in the analysis of heat transfer and diffusion processes.

5. Can g(x,t) and Jn(x) be used to solve any problem involving two variables?

No, g(x,t) and Jn(x) are specific mathematical functions that are useful for solving certain types of problems, particularly those related to waves and vibrations. They may not be applicable to all problems involving two variables, but they are powerful tools in the fields of physics and engineering.

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