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# 00 - Modied Bessel Functions We shall develop the theory of...

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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 > 0 1 a ! ( t 2 ) a u a X b > 0 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 > 0 ( t 2 ) 2 k + n k !( k + n )! = ( t 2 ) n X k > 0 ( 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 0 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

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and therefore 2 I 0 n ( t ) = I n - 1 ( t ) + I n +1 ( t ) . By u : ∂f ∂u = 1 2 t (1 - u - 2 ) f whence X n Z I n ( t ) nu n - 1 = 1 2 t X n Z ( I n ( t ) u n - I n ( t ) u n - 2 ) = 1 2 t X n Z ( I n - 1 ( t ) - I n +1 ( t )) u n - 1 and therefore 2 nI n ( t ) = t
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00 - Modied Bessel Functions We shall develop the theory of...

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