Abstract
We study the distribution of zeros of general solutions of the Airy and Bessel equations in the complex plane. Our results characterize the patterns followed by the zeros for any solution, in such a way that if one zero is known it is possible to determine the location of the rest of zeros.
Highlights
Airy and Bessel functions are important examples of special functions satisfying second order lineal ODEs
We begin with a description of the anti-Stokes lines (ASLs) and Stokes line (SL), followed by an analysis of the distribution of the zeros for the general solution cos αAi(z) + sin αBi(z); this analysis will be important in the description of the zeros of general Bessel functions cos αJν(z) − sin αYν (z)
Differently from the Airy case, asymptotics for large z will be not enough to obtain a complete picture of the distribution of zeros, and we will need to consider uniform asymptotic approximations for large order [9] in order to describe some of the zeros
Summary
Airy and Bessel functions are important examples of special functions satisfying second order lineal ODEs. The use of asymptotics (both of Poincare type and uniform asymptotics for large orders in the case of Bessel functions) will provide more detailed and quantitative information These results will characterize the possible patterns followed by the zeros for any solution, leading to the development of methods which are able to compute safely and accurately all the zeros in a given region and having as only input data the location of just one zero (and the order for the case of the Bessel equation). These are the so-called anti-Stokes lines (ASLs) in the LG approximation. For the zeros of y′(z), as we will see, the approximation that they are on ASL curves given by (2) (and equivalently 4) will be in general sufficiently accurate. The numerical method for complex zeros of y(z) given in [14] is successful for the zeros of y′(z) with only a simple change in the iteration function of the fixed point method
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