High-precision evaluation of the Bessel functions via Hadamard series

We present a method of high-precision calculation of the Bessel functions using Hadamard series. Such series are absolutely convergent expansions involving the normalised incomplete gamma function P ( a , z ) = γ ( a , z ) / Γ ( a ) and possess early terms that behave like those in an asymptotic expansion. In the case of real variables the function P ( a , z ) acts as a smoothing factor on the terms of the series. We show how these series representing the Bessel functions of complex argument can be chosen so as to produce rapidly convergent series that possess terms decaying at the geometric rate ϑ k , where 0 < ϑ < 1 and k is the ordinal number of the series. We give numerical examples with ϑ = 1 2 , 1 3 and 1 4 . The theory is extended to cover the confluent hypergeometric functions F 1 1 ( a ; b ; z ) and U ( a , b , z ) , thereby dealing with many of the special functions arising in mathematical physics.