Question

\((d / d x) J_{0}(x)=-J_{1}(x)\)

Short answer

\(\frac{d}{dx} J_{0}(x) = -J_{1}(x) \)

Step-by-step solution

Understand the given function

The given function is the Bessel function of the first kind, denoted as \(J_{0}(x)\).

Know the properties of the Bessel functions

One important property of Bessel functions is their derivative relationship. Specifically, the derivative of \(J_{0}(x)\) with respect to \(x\) is given by \((d / d x) J_{0}(x)=-J_{1}(x)\).

Differentiate the Bessel function

Apply differentiation to the Bessel function \(J_{0}(x)\). From the given property, we have: \(\frac{d}{dx} J_{0}(x) = -J_{1}(x)\).

Interpret the result

The differentiation result \((d / d x) J_{0}(x)=-J_{1}(x)\) represents a crucial relationship between the first two kinds of Bessel functions.

Key concepts

Bessel function of the first kind

Bessel functions are a family of solutions to Bessel's differential equation. They appear in various physical phenomena, such as heat conduction in cylindrical objects and wave propagation in cylindrical coordinates. The Bessel function of the first kind, denoted as \(J_{n}(x)\), is often encountered in problems with boundary conditions.
For example, \(J_{0}(x)\) is the Bessel function of the first kind of order zero. It looks like a wavy curve that oscillates but eventually diminishes as \(x\) increases.
You typically find these functions when dealing with problems involving cylindrical or spherical symmetry. Using them allows us to find solutions to complex differential equations involving such symmetries.

Derivative Relationship

One of the significant properties of Bessel functions is their derivative relationship. The derivative of the Bessel function of the first kind of order zero, \(J_{0}(x)\), connects directly to the Bessel function of the first kind of order one, \(J_{1}(x)\).
Specifically, the relationship can be stated as:
\( \frac{d}{dx} J_{0}(x) = -J_{1}(x). \)
This means that when you take the derivative of \(J_{0}(x)\) with respect to \(x\), it equals the negative of \(J_{1}(x)\). This derivative relationship is critical because it simplifies complex calculations in various applications.

Differentiation of Bessel Functions

Understanding how to differentiate Bessel functions is crucial for solving differential equations in physics and engineering. The differentiation process is pretty straightforward if you know the specific property.
For instance, given the function \(J_{0}(x)\), applying differentiation using the known property: \( \frac{d}{dx} J_{0}(x) = -J_{1}(x) \)
allows us to simplify our work. This property tells us that the derivative of the Bessel function of the first kind of order zero results in the negative of the Bessel function of the first kind of order one. Being familiar with such properties saves time and avoids potential errors.

Properties of Bessel Functions

Apart from the derivative relationship, Bessel functions have other interesting properties:
- Orthogonality: Bessel functions of different orders are orthogonal to each other under certain integrals. This property is beneficial when solving problems with boundary conditions.
- Recurrence Relations: Bessel functions satisfy recurrence relations that link different orders to each other. For instance, the recurrence relations are: \[ J_{n-1}(x) + J_{n+1}(x) = \frac{2n}{x} J_{n}(x), \ x J_{n-1}(x) - J_{n+1}(x) = 2 J_{n}'(x). \]
- Zeros: The zeros of Bessel functions are the values of \(x\) for which the function equals zero. These zeros are significant in solving partial differential equations with boundary conditions.
Knowing these properties can help understand the behavior of Bessel functions deeply, aiding in their application in solving complex problems.