Bessel functions: Difference between revisions

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In addition, a linear combination of these solutions is also a solution:
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)
(iii) H<sub>&alpha;</sub>(x) = 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.
where C<sub>1</sub> and  C<sub>2</sub> are constants.

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Bessel functions are solutions of the Bessel differential equation:[1][2][3]

where α is a constant.

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

(i) Jα(x) and
(ii) Yα(x).

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

(iii) Hα(x) = C1 Jα(x) + C2 Yα(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.

Applications

Bessel functions arise in many applications. For example, Kepler’s Equation of Elliptical Motion, the vibrations of a membrane, and heat conduction, to name a few.

References

  1. Frank Bowman (1958). Introduction to Bessel Functions, 1st Edition. Dover Publications. ISBN 0-486-60462-4. 
  2. George Neville Watson (1966). A Treatise on the Theory of Bessel Functions, 2nd Edition. Cambridge University Press. 
  3. Bessel Function of the First Kind Eric W. Weisstein, From the website of "MathWorld--A Wolfram Web Resource".