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Bessel's Differential Equations

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Recurrence Relations

Recurrence relations are useful for relating Bessel functions of different orders, and can simplify computation and analysis.

Definition of Bessel's Differential Equation

A Bessel's differential equation is of the form

x^2y'' + xy' + (x^2 - n^2)y = 0

where

n

is a real number known as the order of the Bessel function.

Asymptotic Behavior

As $x$ approaches infinity, Bessel functions exhibit oscillatory decay, which is characteristic of their asymptotic behavior.

Boundary Conditions and Bessel Functions

Proper application of boundary conditions will determine the constants in the general solution of Bessel's differential equation.

Solving for integer $n$

When $n$ is an integer, the general solution is a combination of Bessel functions of the first kind $J_n(x)$ and second kind $Y_n(x)$ , which are linearly independent.

Modified Bessel's Equation

The modified Bessel's equation is like the standard version but with a key difference: $x^2y'' + xy' - (x^2 + n^2)y = 0$ ; it describes exponentially growing and decaying functions.

Solving for $n = 0$

For $n=0$ , the Bessel's equation simplifies, yielding solutions known as Bessel functions of the first and second kind, $J_0(x)$ and $Y_0(x)$ respectively.

Solving for non-integer $n$

For non-integer $n$ , solutions involve Bessel functions of the first kind for both $n$ and $-n$ , $J_n(x)$ and $J_{-n}(x)$ , provided $n$ is not an integer.

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