Is there a shortcut to summing Bessel functions with imaginary units?

In summary, the easiest way to summate the given series is to split it into two sums, one over 2k and one over 2k+1, and use well-known sums involving sine and Bessel functions to find the final solution. This method avoids the need to write out explicit Bessel functions and uses the properties of imaginary unit i to simplify the calculation.
  • #1
matematikuvol
192
0

Homework Statement


What is easiest way to summate
[tex]\sum^{\infty}_{n=1}J_n(x)[i^n+(-1)^ni^{-n}][/tex]
where ##i## is imaginary unit.


Homework Equations





The Attempt at a Solution


I don't need to write explicit Bessel function so in sum could stay
[tex]C_1J_(x)+C_2J_2(x)+...[/tex]
Well I see that terms in the sum will be
[tex]2iJ_1(x)-2J_2(x)+...[/tex]
But I search for more sofisticated solution. Is there any way to sum this using ##i^1=i##,##i^2=-1##,##i^3=-i##,##i^4=1##?
 
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  • #2
cute

first i^n+(-1)^n i^-n=2 i^n

then split into sums over

2k
2k+1

Which have nice well know sums involving sin(x) and J0(x)
 

Related to Is there a shortcut to summing Bessel functions with imaginary units?

1. What is a Bessel function summation?

A Bessel function summation is a mathematical technique used to approximate the value of a Bessel function, which is a special type of mathematical function that arises in many areas of physics and engineering.

2. Why are Bessel function summations important?

Bessel function summations are important because they allow us to estimate the values of Bessel functions, which are used to describe a wide range of physical phenomena, including wave propagation, heat transfer, and quantum mechanics.

3. How is a Bessel function summation calculated?

A Bessel function summation is typically calculated using a series of approximations, such as the partial sum or truncated series methods. These methods involve adding a finite number of terms from the infinite series representation of the Bessel function to obtain an approximate value.

4. What are some practical applications of Bessel function summations?

Bessel function summations have many practical applications in science and engineering. They are commonly used in the analysis of electromagnetic fields, heat transfer, and the design of electronic circuits. They also play a crucial role in solving differential equations that arise in various physical systems.

5. Are there any limitations to using Bessel function summations?

One limitation of using Bessel function summations is that they can only provide approximate values of Bessel functions, which may not be accurate enough for certain applications. Additionally, the convergence of the summation can be slow for some values of the parameters, making it computationally expensive. Other methods, such as numerical integration, may be more suitable in these cases.

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