# Recurrence relations define solutions to Bessel equation

• Dominic Chang
In summary, the conversation discusses the use of recurrence relations to show that a given function satisfies the Bessel differential equation. The individual is having trouble with the substitution process and is seeking clarification on the correct result. They are advised to apply d/dx to the first equation and use the recursion to change all terms to m, which should result in the Bessel equation.
Dominic Chang
I'm trying to show that a function defined with the following recurence relations
$$\frac{dZ_m(x)}{dx}=\frac{1}{2}(Z_{m-1}-Z_{m+1})$$ and $$\frac{2m}{x}Z_m=Z_{m+1}+Z_{m-1}$$ satisfies the Bessel differential equation
$$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$
However, whenever I do the substitution, I end up with
$$-\frac{1}{4m}(Z_{m+2}-Z_{m-2})\neq0$$
I know that these recurence relations can be derived from the solutions to Bessel's equation, so I don't see why I'm not getting the correct result.

Apply d/dx to the first equation to get the second derivative you need. Add 1/x times the first derivative and the second derivative. Use the recursion to change all the m+2, m+1, m-1, and m-2 into m. Collect like terms. You should get Bessel.

## 1. What is a recurrence relation?

A recurrence relation is a mathematical equation that describes the relationship between one term of a sequence and the previous terms. It is used to generate a sequence of values that depend on the previous values.

## 2. What is the Bessel equation?

The Bessel equation is a second-order differential equation that arises in many areas of physics and engineering. It is named after the mathematician Friedrich Bessel and is used to model wave phenomena such as sound and light.

## 3. How are recurrence relations used to solve the Bessel equation?

Recurrence relations can be used to find solutions to the Bessel equation by expressing the solution as a power series. The coefficients of the power series can be determined by using the recurrence relation, and the series can be truncated to obtain an approximate solution.

## 4. What are the applications of Bessel equations and recurrence relations?

Bessel equations and recurrence relations have many applications in physics and engineering, including the analysis of wave phenomena, heat transfer, and quantum mechanics. They are also used in signal processing and image analysis.

## 5. Are there any limitations to using recurrence relations to solve the Bessel equation?

While recurrence relations can provide solutions to the Bessel equation, they may not always be accurate or efficient. In some cases, other methods such as numerical methods or special functions may be needed to obtain more accurate solutions.

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