Definition:Bessel's Equation

Equation

Bessel's equation is a second order ODE of the form:

$x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} + \paren {x^2 - n^2} y = 0$

The parameter $n$ may be any arbitrary real or complex number.

Solution

The solutions of Bessel's equation with parameter $n$ are known as Bessel functions of order $n$, and they are functions of the parameter $n$.

Also presented as

Some sources give Bessel's equation as:

$x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} + \paren {\lambda^2 x^2 - n^2} y = 0$

Also known as

Bessel's equation is also referred to as Bessel's differential equation.

The parameter $n$ is variously presented. Some sources use $p$.

Also see

• Definition:Bessel's Modified Equation

• Results about Bessel's equations can be found here.

Source of Name

This entry was named for Friedrich Wilhelm Bessel.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {II}$. Bessel functions: $5$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 24$: Bessel Functions: Bessel's Differential Equation: $24.1$
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction: $(9)$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Bessel's equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bessel's equation
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Bessel's equation