Bessel functions: Difference between revisions

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Bessel functions are solutions of the Bessel differential equation:

$z2(d2w/dz2)+z(dw/dz)+(z2-alpha2)$

where alpha is a constant.

Because this is a second-order differential equation, it should have two linearly-independent solutions:

(i) J_alpha(x) and (ii) Y_alpha(x).

In addition, a linear combination of these solutions is also a solution:

(iii) H_alpha = C₁ J_alpha(x) + C₂ Y_alpha(x)

where C₁ and C₂ are constants.

These three kinds of solutions are called Bessel functions of the first kind, second kind, and third kind.

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+Bessel functions are solutions of the Bessel differential equation:
+<math>z<sup>2</sup> (d<sup>2</sup>w/dz<sup>2</sup>) + z (dw/dz) + (z<sup>2</sup> - alpha<sup>2</sup>)</math>
+where alpha is a constant.
+Because this is a second-order differential equation, it should have two linearly-independent solutions:
+(i) J<sub>alpha</sub>(x) and
+(ii) Y<sub>alpha</sub>(x).
+In addition, a linear combination of these solutions is also a solution:
+(iii) H<sub>alpha</sub> = C<sub>1</sub> J<sub>alpha</sub>(x) + C<sub>2</sub> Y<sub>alpha</sub>(x)
+where C<sub>1</sub> and  C<sub>2</sub> are constants.
+These three kinds of solutions are called Bessel functions of the first kind, second kind, and third kind.