00 - Modified Bessel Functions We shall develop the theory...

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Unformatted text preview: Modified Bessel Functions We shall develop the theory of modified Bessel functions from their gen- erating function. Explicitly, define f ( t,u ) = exp { 1 2 t ( u + u- 1 ) } for t and u 6 = 0 real (though we may also allow complex when appropriate). Note that if in the (absolutely convergent) expansion f ( t,u ) = X a > 1 a ! ( t 2 ) a u a X b > 1 b ! ( t 2 ) b u- b we collect together terms according to powers of u then we obtain f ( t,u ) = X n Z I n ( t ) u n where (letting k = b and a = b + n ) if n > 0 then I- n = I n and I n ( t ) = X k > ( t 2 ) 2 k + n k !( k + n )! = ( t 2 ) n X k > ( t 2 ) 2 k k !( k + n )! is the modified Bessel function of order n . Differentiation of f ( t,u ) by t and u yields relations between consecutive modified Bessel functions and their derivatives. By t : f t = 1 2 ( u + u- 1 ) f whence X n Z I n ( t ) u n = 1 2 X n Z ( I n ( t ) u n +1 + I n ( t ) u n- 1 ) = X n Z ( I n- 1 ( t ) + I n +1 ( t )) u n 1 and therefore 2 I n ( t ) =...
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.

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00 - Modified Bessel Functions We shall develop the theory...

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