• xylai
In summary, the conversation discusses the Bessel function and its relationship with its negative order. It is mentioned that when the order is an integer, the relationship is straightforward, but when it is not an integer, it becomes more complicated. The formula for the relationship is also mentioned, which involves cosine and sine functions. The use of Numerical Recipes is suggested for obtaining the desired values.
xylai
I am working on some numerical works. I use the computer language: Fortran language.
Here I have a problem about the Bessel functon.

Now I know the value of Bessel[v,x], where v is positive and real.
I want to know the value of Bessel[-v,x].

I don't know their relation. Can you help me?
Thanks!

The Bessel function satisfies the differential equation,

$$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \nu^2)y = 0$$

We can see here that the sign of the order wold seem to be irrelevant because we take its square. However, the relationship is

$$J_{-n}(x) = (-1)^nJ_n(x)$$

Born2bwire said:
The Bessel function satisfies the differential equation,

$$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \nu^2)y = 0$$

We can see here that the sign of the order wold seem to be irrelevant because we take its square. However, the relationship is

$$J_{-n}(x) = (-1)^nJ_n(x)$$

As far as I know, when n is integer, you are right: $$J_{-n}(x) = (-1)^nJ_n(x)$$.
But when n is not integer, it becomes very difficult.

xylai said:
As far as I know, when n is integer, you are right: $$J_{-n}(x) = (-1)^nJ_n(x)$$.
But when n is not integer, it becomes very difficult.

$$J_{-\nu} =\cos (\nu\pi)J_\nu - \sin(\nu\pi)Y_\nu$$

via Numerical Recipes.

You are very clever.

## 1. What is a Bessel function?

A Bessel function is a type of special function that arises in mathematical physics, particularly in the study of wave phenomena. It was first introduced by the German mathematician Friedrich Bessel in the early 19th century.

## 2. What are the applications of Bessel functions?

Bessel functions have a wide range of applications in physics and engineering, including solving partial differential equations, describing the behavior of electromagnetic waves, and modeling the vibrations of circular membranes and disks.

## 3. How are Bessel functions different from other special functions?

Bessel functions are unique because they are solutions to a special type of differential equation known as Bessel's equation. They also have a wide range of properties and applications that make them distinct from other special functions, such as trigonometric functions and hypergeometric functions.

## 4. Can Bessel functions be used in real-world problems?

Yes, Bessel functions have many practical applications in various fields, including acoustics, optics, electromagnetics, and signal processing. They can be used to model and analyze physical phenomena and provide solutions to engineering problems.

## 5. Can Bessel functions be computed efficiently?

Yes, there are various algorithms and numerical methods that have been developed to compute Bessel functions efficiently. These include recurrence relations, power series expansions, and asymptotic approximations. Additionally, many mathematical software packages have built-in functions for computing Bessel functions.

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