Definition:Bessel's Modified Equation

Equation

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

Also known as

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

Some sources present it as the modified Bessel equation.

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

Also see

• Definition:Bessel's Equation
• Definition:Modified Bessel Function

• Results about Bessel's modified equation can be found here.

Source of Name

This entry was named for Friedrich Wilhelm Bessel.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 24$: Bessel Functions: Bessel's Modified Differential Equation: $24.31$
• 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