scholarly journals On the complex zeros of Airy and Bessel functions and those of their derivatives

2014 ◽  
Vol 12 (05) ◽  
pp. 537-561 ◽  
Author(s):  
Amparo Gil ◽  
Javier Segura
Keyword(s):  
Complex Plane ◽  
Bessel Functions ◽  
Distribution Of Zeros ◽  
General Solutions ◽  
Complex Zeros

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.

Innovative Genetic Algorithmic Approach to Select Potential Patches Enclosing Real and Complex Zeros of Nonlinear Equation

2017 ◽  
Vol 6 (2) ◽  
pp. 18-37 ◽  
Author(s):  
Vijaya Lakshmi V. Nadimpalli ◽  
Rajeev Wankar ◽  
Raghavendra Rao Chillarige
Keyword(s):  
Genetic Algorithm ◽  
Nonlinear Equation ◽  
Complex Plane ◽  
Search Space ◽  
Regions Of Interest ◽  
Real Numbers ◽  
Algorithmic Approach ◽  
Space Reduction ◽  
Complex Zeros ◽  
The Given

In this article, an innovative Genetic Algorithm is proposed to find potential patches enclosing roots of real valued function f:R→R. As roots of f can be real as well as complex, the function is reframed on to complex plane by writing it as f(z). Thus, the problem now is transformed to finding potential patches (rectangles in C) enclosing z such that f(z)=0, which is resolved into two components as real and imaginary parts. The proposed GA generates two random populations of real numbers for the real and imaginary parts in the given regions of interest and no other initial guesses are needed. This is the prominent advantage of the method in contrast to various other methods. Additionally, the proposed ‘Refinement technique' aids in the exhaustive coverage of potential patches enclosing roots and reinforces the selected potential rectangles to be narrow, resulting in significant search space reduction. The method works efficiently even when the roots are closely packed. A set of benchmark functions are presented and the results show the effectiveness and robustness of the new method.

The phase problem in scattering phenomena: the zeros of entire functions and their significance

1978 ◽  
Vol 360 (1700) ◽  
pp. 25-45 ◽  
Keyword(s):  
Entire Functions ◽  
Signal To Noise Ratio ◽  
Phase Problem ◽  
Distribution Of Zeros ◽  
Real Zeros ◽  
Zeros Of Entire Functions ◽  
Functions Of Exponential Type ◽  
Zeros Of Functions ◽  
The Fourier Transform ◽  
Complex Zeros

The paper outlines an approach to the calculation of the phase from intensity data based on the properties of the distribution of zeros of functions of exponential type. This leads to a reinterpretation of such phenomena as Gibbs’ or speckle, which underlines their intrinsic unity. The phase problem is solved for functions which present complex zeros by apodization, i.e. by creating a sufficiently large zero-free area. The method is based on a compromise between signal to noise ratio and resolution and is meaningful provided the apodization required is not too severe. Real zeros, for which the phase problem is trivial, occur only for the special case of eigenfunctions of the Fourier transform

The distribution of zeros of spherical bessel functions

1989 ◽  
Vol 103 (6) ◽  
pp. 611-616 ◽  
Author(s):  
E. R. Arriola ◽  
J. S. Dehesa
Keyword(s):  
Bessel Functions ◽  
Distribution Of Zeros

The asymptotic expansion of bessel functions of large order

1954 ◽  
Vol 247 (930) ◽  
pp. 328-368 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Large Order ◽  
Hankel Functions ◽  
Rapid Calculation ◽  
Zeros Of Functions ◽  
Significant Figures ◽  
Complex Zeros

New expansions are obtained for the functions Iv{yz), ) and their derivatives in terms of elementary functions, and for the functions J v(vz), Yv{vz), H fvz) and their derivatives in terms of Airy functions, which are uniformly valid with respect to z when | | is large. New series for the zeros and associated values are derived by reversion and used to determine the distribution of the zeros of functions of large order in the z-plane. Particular attention is paid to the complex zeros of 7„(z) and the Hankel functions when the order n is an integer or half an odd integer, and for this purpose some new asymptotic expansions of the Airy functions are derived. Tables are given of complex zeros of Airy functions and other quantities which facilitate the rapid calculation of the smaller complex zeros of 7„(z), 7'(z), and the Hankel functions and their derivatives, when 2 n is an integer, to an accuracy of three or four significant figures.

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