Derivative of Bessel Function

We now explore the derivative of Bessel functions, commonly represented as d/dx [J_n(x)] or [J_n(x)]'. Bessel functions, particularly of the first kind, have well-defined derivatives, indicating differentiability within their domain.

Key concepts include: -

Bessel Function: J_n(x) is a solution to Bessel's differential equation. 

Chain Rule: Useful for differentiating compositions involving Bessel functions. 

Recurrence Relations: Used in deriving the derivatives of Bessel functions.

The derivative of a Bessel function of the first kind, J_n(x), can be denoted as d/dx [J_n(x)] or [J_n(x)]'.

The formula is: d/dx [J_n(x)] = (J_{n-1}(x) - J_{n+1}(x))/2

This formula applies to all x where J_n(x) is defined, using the recurrence relations of Bessel functions.

We can derive the derivative of Bessel functions using several methods, leveraging their recurrence relations and properties.

Methods include: 

1. Using Recurrence Relations 
2. Application of Chain Rule -

Direct Differentiation of Bessel's Differential Equation We will now demonstrate that the differentiation of J_n(x) results in (J_{n-1}(x) - J_{n+1}(x))/2

using these methods:

Using Recurrence Relations

The derivative of J_n(x) can be derived from the recurrence relations of Bessel functions.

Consider the recurrence relations: J_{n-1}(x) + J_{n+1}(x) = (2n/x)J_n(x)

Differentiating both sides with respect to x, we get: d/dx [J_{n-1}(x)] + d/dx [J_{n+1}(x)] = (2n/x^2)J_n(x) - (2n/x)d/dx [J_n(x)]

Rearranging gives: d/dx [J_n(x)] = (J_{n-1}(x) - J_{n+1}(x))/2

Using Chain Rule

Consider J_n(x) as a composition of functions; differentiate using the chain rule: If J_n(x) is expressed in terms of simpler functions, apply the chain rule accordingly.

For instance, if J_n(x) = f(g(x)), then: d/dx [J_n(x)] = f'(g(x))g'(x)

Direct Differentiation of Bessel's Differential Equation Bessel's differential equation is: x^2y'' + xy' + (x^2 - n^2)y = 0

Differentiating directly with respect to x, we can solve for y' to find the derivative of J_n(x).

When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives. For Bessel functions, these derivatives are essential in applications like solving boundary value problems.

For the first derivative of J_n(x), we write f′(x), indicating how the function changes at a point.

The second derivative is derived from the first, denoted as f′′(x), and so on.

For the nth derivative of J_n(x), we generally use f^(n)(x) to indicate the change in the rate of change.

Bessel functions have unique properties at certain points: - At x = 0, the derivative of J_n(x) may be undefined depending on n. 

For large x, asymptotic expansions help approximate the derivatives of Bessel functions.

• Bessel Function: A solution to Bessel's differential equation, often used in physics and engineering.

• Derivative: Indicates how a function changes in response to a slight change in its argument.

• Recurrence Relation: A relation that expresses Bessel functions in terms of other Bessel functions of different orders.

• Chain Rule: A rule for differentiating compositions of functions.

• Quotient Rule: A technique for differentiating ratios of functions.