Abstract
We present accurate trigonometric expansions of Bessel functions of first kind and integer order for complex arguments of the form <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Jv</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> )=Σ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> α <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kS</i> (β <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kz</i> ), where α <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> and β <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> are constants and S is a sinusoidal function. Using the new expansions, varying levels of accuracy and range of applicability can be achieved by varying the number of terms in the expansions. For example, a four term expansion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> ) yields an average relative error of <; .1% for | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> | ≤ 2π and same accuracy is achieved for an eight term expansion for an extended range |z| ≤ 5π. Further, a phase and amplitude corrected large argument asymptotic formula is studied such that, the lower limit of its usage is reduced to medium magnitude ranges of arguments. The new set of formulas can not only be incorporated into math libraries very easily but also be useful for treatment of radiation and scattering problems involving Bessel functions.
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