Bessel function of the first kind with complex argument |
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Authors: | Hashim A Yousif Richard Melka |
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Affiliation: | Natural Sciences Division, University of Pittsburgh, 300 Campus Drive, Bradford, PA 16701, USA |
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Abstract: | A new method of computing integral order Bessel functions of the first kind Jn(z) when either the absolute value of the real part or the imaginary part of the argument z = x + iy is small, is described. This method is based on computing the Bessel functions from asymptotic expressions when x∼ 0 (or y ∼ 0). These expansions are derived from the integral definition of Bessel functions. This method is necessary because some existing algorithms and methods fail to give correct results for small x small y. In addition, our overall method of computing Bessel functions of any order and argument is discussed and the logarithmic derivative is used in computing these functions. The starting point of the backward recurrence relations needed to evaluate the Bessel function and their logarithmic derivatives are investigated in order to obtain accurate numerical results. Our numerical method, together with established techniques of computing the Bessel functions, is easy to implement, efficient, and produces reliable results for all z. |
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Keywords: | Cylindrical Bessel functions Jn(z) Bessel functions Logarithmic derivative of Jn(z) Numerical evaluation of Jn(z) Asymptotic expansion of Jn(z) |
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