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Zeros of the Macdonald function of complex order. (English) Zbl 1132.33309
Summary: The $z$-zeros of the modified Bessel function of the third kind $K_{\nu }(z)$, also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order $\nu $. Approximate expressions for the zeros, applicable in the cases of very small or very large $|\nu |$, are given. The behaviour of the zeros for varying $|\nu |$ or arg$\nu $, obtained numerically, is illustrated by means of some graphics.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
Algorithm 831
Full Text: DOI arXiv
[1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.
[2] Campbell, J. B.: On temme’s algorithm for the modified Bessel function of the third kind, ACM trans. Math. software 6, 581-586 (1980) · Zbl 0447.33005 · doi:10.1145/355921.355928
[3] Cochran, J. A.: The zeros of Hankel functions as functions of their order, Numer. math. 7, 238-250 (1965) · Zbl 0145.07503 · doi:10.1007/BF01436080
[4] Cochran, J. A.; Hoffspiegel, J. N.: Numerical techniques for finding $\nu $-zeros of Hankel functions, Math. comput. 24, 413-422 (1970) · Zbl 0219.65049 · doi:10.2307/2004488
[5] Cruz, A.; Esparza, J.; Sesma, J.: Zeros of the Hankel function of real order out of the principal Riemann sheet, J. comput. Appl. math. 37, 89-99 (1991) · Zbl 0743.33006 · doi:10.1016/0377-0427(91)90109-W
[6] Cruz, A.; Sesma, J.: Zeros of the Hankel function of real order and of its derivative, Math. comput. 39, 639-645 (1982) · Zbl 0511.33002 · doi:10.2307/2007340
[7] Dunster, T. M.: Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter, SIAM J. Math. anal. 21, 995-1018 (1990) · Zbl 0703.33002 · doi:10.1137/0521055
[8] Fabijonas, B. R.; Lozier, D. W.; Rappoport, J. M.: Algorithms and codes for the macdonald function: recent progress and comparisons, J. comput. Appl. math. 161, 179-192 (2003) · Zbl 1033.65010 · doi:10.1016/S0377-0427(03)00596-X
[9] Fabijonas, B. R.; Olver, F. W. J.: On the reversion of an asymptotic expansion and the zeros of the Airy functions, SIAM rev. 41, 762-773 (1999) · Zbl 1053.33003 · doi:10.1137/S0036144598349538
[10] Ferreira, E. M.; Sesma, J.: Zeros of the modified Hankel function, Numer. math. 16, 278-284 (1970) · Zbl 0219.65048 · doi:10.1007/BF02219779
[11] Franz, W.; Galle, R.: Semiasymptotische reihen für die beugung einer ebenen welle am zylinder, Z. naturforschg. 10a, 374-378 (1955) · Zbl 0064.44101
[12] Friedlander, F. G.: Diffraction of pulses by a circular cylinder, Commun. pure appl. Math. 7, 705-732 (1954) · Zbl 0057.18701 · doi:10.1002/cpa.3160070407
[13] Gautschi, W.: Numerical quadrature computation of the macdonald function for complex orders, BIT numer. Math. 45, 593-603 (2005) · Zbl 1082.65026 · doi:10.1007/s10543-005-0020-5
[14] Gil, A.; Segura, J.; Temme, N. M.: Evaluation of the modified Bessel function of the third kind of imaginary orders, J. comput. Phys. 175, 398-411 (2002) · Zbl 0996.65026 · doi:10.1006/jcph.2001.6894
[15] Gil, A.; Segura, J.; Temme, N. M.: Computing special functions by using quadrature rules, Numer. algorithms 33, 265-275 (2003) · Zbl 1031.65033 · doi:10.1023/A:1025524324969
[16] Gil, A.; Segura, J.; Temme, N. M.: Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments, ACM trans. Math. software 30, 145-158 (2004) · Zbl 1072.65024 · doi:10.1145/992200.992203
[17] Gil, A.; Segura, J.; Temme, N. M.: Algorithm 831: modified Bessel functions of imaginary order and positive argument, ACM trans. Math. software 30, 159-164 (2004) · Zbl 1072.65025 · doi:10.1145/992200.992204
[18] Gray, A.; Mathews, G. B.: A treatise on Bessel functions and their application to physics, (1966) · Zbl 0135.28002
[19] Hethcote, H. W.: Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems, J. math. Phys. 11, 2501-2504 (1970) · Zbl 0195.34801 · doi:10.1063/1.1665417
[20] Keller, J. B.; Rubinow, S. I.; Goldstein, M.: Zeros of Hankel functions and poles of scattering amplitudes, J. math. Phys. 4, 829-832 (1963) · Zbl 0115.05903 · doi:10.1063/1.1724325
[21] Laforgia, A.: Inequalities and monotonicity results for zeros of modified Bessel functions of purely imaginary order, Quart. appl. Math. 44, 91-96 (1986) · Zbl 0597.33006
[22] Leung, K. V.; Ghaderpanah, S. S.: An application of the finite element approximation method to find the complex zeros of the modified Bessel function $Kn(z)$, Math. comput. 33, 1299-1306 (1979) · Zbl 0427.30009 · doi:10.2307/2006465
[23] Lozier, D. W.: Software needs in special functions, J. comput. Appl. math. 66, 345-358 (1996) · Zbl 0855.65009 · doi:10.1016/0377-0427(95)00181-6
[24] Lozier, D. W.; Olver, F. W. J.: Numerical evaluation of special functions, Mathematics of computation 1943 -- 1993: A half-century of computational mathematics, proc. Symposia in applied mathematics 48, 79-125 (1994) · Zbl 0815.65030
[25] Magnus, W.; Kotin, L.: The zeros of the Hankel function as a function of its order, Numer. math. 2, 228-244 (1960) · Zbl 0095.05303 · doi:10.1007/BF01386226
[26] Nalesso, G. F.: On the zeros of a class of Bessel functions whose argument and order are functions of a complex variable, IMA J. Appl. math. 43, 195-217 (1989) · Zbl 0711.33002 · doi:10.1093/imamat/43.3.195
[27] S.E. Sandström, C. Ackrén, Note on the complex zeros of H\nu $^{\prime}$(x)+i\zeta H\nu (x)=0, J. Comput. Appl. Math., in press. Available online, DOI:10.1016/j.cam.2006.01.032.
[28] Segura, J.: The zeros of special functions from a fixed point method, SIAM J. Numer. anal. 40, 114-133 (2002) · Zbl 1058.33020 · doi:10.1137/S0036142901387385
[29] Segura, J.; De Córdoba, P. Fernández; Ratis, Yu.L.: A code to evaluate modified Bessel functions based on the continued fraction method, Comput. phys. Commun. 105, 263-272 (1997) · Zbl 0930.65009 · doi:10.1016/S0010-4655(97)00069-6
[30] Temme, N. M.: On the numerical evaluation of the modified Bessel function of the third kind, J. comput. Phys. 19, 324-337 (1975) · Zbl 0334.65013 · doi:10.1016/0021-9991(75)90082-0
[31] Temme, N. M.: Numerical algorithms for uniform Airy-type asymptotic expansions, Numer. algorithms 15, 207-225 (1997) · Zbl 0886.65012 · doi:10.1023/A:1019197921337
[32] Thompson, I. J.; Barnett, A. R.: Modified Bessel functions I$\nu (z)$ and K$\nu (z)$ of real order and complex argument, to selected accuracy, Comput. phys. Commun. 47, 245-257 (1987)
[33] Watson, G. N.: A treatise on the theory of Bessel functions, (1944) · Zbl 0063.08184