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Unformatted text preview: USEFUL FACTS AND FORMULAS FOR BESSEL FUNCTIONS 1. Bessel equation Bessel functions arise from solving Laplace's equation in cylindrical co-ordinates. Bessel functions of order v are solutions of the Bessel equation 2 2 2 2 1 1 0. d R dR R dx x dx x + +- = (1.1) 2. Bessel functions of the first kind If v 2 0, and v is not an integer, then the two linearly independent solutions are ( ) ( ) ( ) 2 1 , 2 ! 1 2 k k k x x J x k k =- = + + (2.1) and ( ) ( ) ( ) 2 1 . 2 ! 1 2 k k k x x J x k k - - =- = - + (2.2) These solutions are called Bessel functions of the first kind of order v . 3. Bessel functions of the second kind If v = n , an integer, the two solutions (2.1) and (2.2) are linearly dependent, and it is necessary to find another linearly independent solution when v is an integer. It is convention that, even when v is not an integer, the pair of solutions J v ( x ) are replaced by J v ( x ) and N v ( x ) where ( ) ( ) ( ) cos...
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- Fall '11