T. Mark Dunster

Ph.D. Applied Mathematics, University of Bristol, U.K.             

Professor, Department of Mathematics & Statistics
College of Sciences
San Diego State University
5500 Campanile Drive
San Diego, CA 92182-7720
USA

Research Areas: Asymptotic analysis, special functions, ordinary differential equations, scattering theory.

 

Publications 

  1. W. G .C. Boyd and T. M. Dunster, Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. Anal. 17 (2) (1986), 422-450, https://doi.org/10.1137/0517033
  2. T. M. Dunster, Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6) (1986), 1495-1524, https://doi.org/10.1137/0517108
  3. T. M. Dunster, Uniform asymptotic expansions for Whittaker's confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3) (1989), 744-760, https://doi.org/10.1137/0520052
  4. T. M. Dunster, Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4) (1990), 995-1018, https://doi.org/10.1137/0521055
  5. T. M. Dunster, Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6) (1990), 1594-1618, https://doi.org/10.1137/0521087
  6. T. M. Dunster and D. A. Lutz, Convergent factorial series expansions for Bessel functions. SIAM J. Math. Anal. 22 (4) (1991), 1156-1172, https://doi.org/10.1137/0522075
  7. T. M. Dunster, Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sec. A. 119 (3-4) (1991), 311-327, https://doi.org/10.1017/S0308210500014864
  8. T. M. Dunster, Uniform asymptotic expansions for oblate spheroidal functions I: positive separation parameter λ\lambda. Proc. Roy. Soc. Edinburgh Sec. A. 121 (3-4) (1992), 303-320, https://doi.org/10.1017/S0308210500027931
  9. T. M. Dunster, D. A. Lutz and R. Schäfke, Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London, Ser. A 440 (1993), 37-54, https://doi.org/10.1098/rspa.1993.0003
  10. T. M. Dunster, Uniform asymptotic approximations for Mathieu functions. Methods Appl. Anal. 1 (2) (1994), 143-168, https://doi.org/10.4310/MAA.1994.v1.n2.a2
  11. T. M. Dunster, Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal.  25 (2) (1994), 322-353, https://doi.org/10.1137/S0036141092229537
  12. T. M. Dunster, Uniform asymptotic expansions for oblate spheroidal functions II: negative separation parameter λ\lambda\lambda. Proc. Roy. Soc. Edinburgh Sec. A. 125 (4) (1995), 719-737, https://doi.org/10.1017/S0308210500030316
  13. T. M. Dunster, Asymptotics of the generalised exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes' discontinuities. Proc. Roy. Soc. London Ser. A  452 (1996), 1351-1367, https://doi.org/10.1098/rspa.1996.0069
  14. T. M. Dunster, Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete Gamma function. Proc. Roy. Soc. London Ser. A  452 (1996), 1331-1349, https://doi.org/10.1098/rspa.1996.0068
  15. T. M. Dunster, Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1) (1996), 109-134, https://doi.org/10.4310/MAA.1996.v3.n1.a7
  16. T. M. Dunster, Error analysis in a uniform asymptotic expansion for the generalised exponential integral. J. Comp. Appl. Math. 80 (1) (1997), 127-161, https://doi.org/10.1016/S0377-0427(97)00026-5
  17. T. M. Dunster, R. B. Paris and S. Cang, On the high-order coefficients in the uniform asymptotic expansion for the incomplete Gamma function. Methods Appl. Anal. 5 (3) (1998), 223-247, https://doi.org/10.4310/MAA.1998.v5.n3.a1
  18. T. M. Dunster, Asymptotics of the eigenvalues of a rotating harmonic oscillator.  J. Comp. Appl. Math. 93 (1) (1998), 45-73, https://doi.org/10.1016/S0377-0427(98)00070-3
  19. T. M. Dunster, Uniform asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3) (1999), 281-316, https://doi.org/10.4310/MAA.1999.v6.n3.a2
  20. T. M. Dunster, Uniform asymptotic expansions for the reverse generalised Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5) (2001), 987-1013, https://doi.org/10.1137/S0036141099359068
  21. T. M. Dunster, Convergent expansions for linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3) (2001), 293-323, https://doi.org/10.1111/1467-9590.00188
  22. T. M. Dunster, Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1) (2001), 93-133, https://doi.org/10.1006/jath.2001.3595
  23. T. M. Dunster, Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sec. A. 133 (4) (2003), 807-827, https://doi.org/10.1017/S0308210500002687
  24. T. M. Dunster, Uniform asymptotic approximations for the Whittaker functions Mκ,iμ(z)M_{\kappa,i\mu}(z) and Wκ,iμ(z)W_{\kappa,i\mu}(z). Anal. Appl. 1 (2) (2003), 199-212, https://doi.org/10.1142/S0219530503000119
  25. T. M. Dunster, Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3) (2004), 245-270, https://doi.org/10.1111/j.0022-2526.2004.01525.x
  26. T. M. Dunster, Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2) (2006), 339-353, https://doi.org/10.1016/j.cam.2004.11.051
  27. T. M. Dunster, M. Yedlin and K. Lam, Resonance and the late coefficients in the scattered field of a dielectric circular cylinder. Anal. Appl. 4 (4) (2006), 311-333, https://doi.org/10.1142/S0219530506000796
  28. T. M. Dunster, On the logarithmic derivative of Nicholson’s integral. Anal. Appl. 7 (1) (2009), 73-86, https://doi.org/10.1142/S0219530509001281
  29. T. M. Dunster, Quasi nonuniqueness in the scattered field of a dielectric circular cylinder. Anal. Appl. 8 (1) (2010), 63-83, https://doi.org/10.1142/S0219530510001515
  30. T. M. Dunster, Simplified asymptotic solutions of differential equations having double turning points, with an application to Legendre functions. Stud. Appl. Math. 127 (3) (2011), 250-283, https://doi.org/10.1111/j.1467-9590.2011.00519.x
  31. T. M. Dunster, Conical functions of purely imaginary order and argument. Proc. Roy. Soc. Edinburgh Sec. A. 143 (5) (2013), 929-955, https://doi.org/10.1017/S0308210511001582
  32. T. M. Dunster, Electromagnetic wave scattering by two parallel infinite dielectric cylinders. Stud. Appl. Math. 131 (3) (2013), 302-316, https://doi.org/10.1111/sapm.12014
  33. T. M. Dunster, Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. 12 (4) (2014), 385-402, https://doi.org/10.1142/S0219530514500298
  34. T. M. Dunster, A. Gil, J. Segura and N. M. Temme, Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders. Numer. Algorithms 68 (2015), 497-509, https://doi.org/10.1007/s11075-014-9857-5
  35. T. M. Dunster, On the order derivatives of Bessel functions. Constr. Approx. 46 (1) (2017), 47-68, https://doi.org/10.1007/s00365-016-9355-1
  36. T. M. Dunster, Asymptotics of prolate spheroidal wave functions. J. Classical Anal. 11 (1) (2017), 1-21, https://doi.org/10.7153/jca-11-01
  37. T. M. Dunster, A. Gil and J. Segura, Computation of asymptotic expansions of turning point problems via Cauchy's integral formula: Bessel functions. Constr. Approx. 46 (3) (2017), 645-675, https://doi.org/10.1007/s00365-017-9372-8
  38. T. M. Dunster, A. Gil, J. Segura and N. M. Temme, Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions. Comput. Phys. Commun. 217 (2017), 193-197, https://doi.org/10.1016/j.cpc.2017.04.007
  39. T. M. Dunster, A. Gil and J. Segura, Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions. Adv. Comput. Math. 44 (5) (2018), 1441-1474, https://doi.org/10.1007/s10444-018-9589-5
  40. H. S. Cohl, T. H. Dang and T. M. Dunster, Fundamental solutions and Gegenbauer expansions of Helmholtz operators on Riemannian spaces of constant curvature. SIGMA 14 (2018), 136, 45 pages. https://doi.org/10.3842/SIGMA.2018.136
  41. T. M. Dunster, Liouville-Green expansions of exponential form, with an application to modified Bessel functions. Proc.  Roy.  Soc.  Edinburgh  Sec.  A. 150 (3) (2020), 1289-1311, https://doi.org/10.1017/prm.2018.117
  42. T. M. Dunster, Asymptotic solutions of inhomogeneous differential equations having a turning point. Stud. Appl. Math. 145 (3) (2020), 500-536, https://doi.org/10.1111/sapm.12326
  43. T. M. Dunster, A. Gil and J. Segura, Simplified error bounds for turning point expansions. Anal. Appl. 19 (4) (2021), 647-678, https://doi.org/10.1142/S0219530520500104
  44. T. M. Dunster, Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations. J. Classical Anal. 17 (1) (2021), 69-107, https://doi.org/10.7153/jca-2021-17-06
  45. T. M. Dunster, A. Gil and J. Segura, Sharp error bounds for turning point expansions. J. Classical Anal.  18 (1) (2021), 49-81, https://doi.org/10.7153/jca-2021-18-05
  46. T. M. Dunster, Nield-Kuznetsov functions and Laplace transforms of parabolic cylinder functions. SIAM J. Math. Anal. 53 (5), (2021) 5915-5947, https://doi.org/10.1137/21M1401590
  47. T. M. Dunster, Uniform asymptotic expansions for the Whittaker functions Mκ,μ(z)M_{\kappa,\mu}(z) and Wκ,μ(z)W_{\kappa,\mu}(z) with Î¼\mu large. Proc. A. 477 (2021), no. 2252, Paper No. 20210360, 18 pp, https://doi.org/10.1098/rspa.2021.0360
  48. T. M. Dunster and J. M. Perez, On the coefficients in an asymptotic expansion of (1+1/x)x . Involve. 14-5 (2021), 775-781, https://doi.org/10.2140/involve.2021.14.775
  49. T. M. Dunster, A. Gil, D. Ruiz-Antolín and J. Segura, Computation of the reverse generalized Bessel polynomials and their zeros. Comput. Math. Methods 3 (6) (2021), Paper No. e1198, 12 pp, https://doi.org/10.1002/cmm4.1198
  50. T. M. Dunster, Uniform asymptotic expansions for Lommel, Anger-Weber, and Struve functions, Stud. Appl. Math. 148 (1) (2022), 340-372, https://doi.org/10.1111/sapm.12442
  51. T. M. Dunster, Uniform asymptotic expansions for Gegenbauer polynomials and related functions via differential equations having a simple pole. Constr. Approx.  (59) (2024), 419-456, https://doi.org/10.1007/s00365-023-09645-1
  52. T. M. Dunster, Uniform asymptotic expansions for the zeros of Bessel functions. SIAM J. Math. Anal. 56 (5) (2024), 6521-6550, https://doi.org/10.1137/23M1611269
  53. T. M. Dunster, A. Gil and J. Segura, Computation of parabolic cylinder functions having complex argument. Appl. Numer. Math. 197 (2024) 230-242, https://doi.org/10.1016/j.apnum.2023.11.017
  54. T. M. Dunster, Error bounds for a uniform asymptotic approximation of the zeros of the Bessel function Jν(x)J_{\nu}(x). Anal. Appl. 23 (8) (2025), 1501-1537, https://doi.org/10.1142/S0219530525500137
  55. T. M. Dunster, Simplified uniform asymptotic expansions for associated Legendre and conical functions. J. Approx. Theory (2025), https://doi.org/10.1016/j.jat.2025.106228
  56. T. M. Dunster, Uniform asymptotic expansions for Bessel functions of imaginary order and their zeros. Integral Transforms Spec. Funct. (2025), http://dx.doi.org/10.1080/10652469.2025.2524721
  57. T. M. Dunster, A. Gil, D. Ruiz-Antolín and J. Segura, A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane. BIT Numer. Math. 65 (20) (2025), https://doi.org/10.1007/s10543-025-01065-w
  58. T. M. Dunster, A. Gil, D. Ruiz-Antolín and J. Segura, Uniform asymptotic expansions for the zeros of parabolic cylinder functions. Stud. Appl. Math. 154 (2025), p. e70004, https://doi.org/10.1111/sapm.70004
  59. T. M. Dunster, Simplified Airy function asymptotic expansions for reverse generalised Bessel polynomials. J. Classical Anal. (2025), Submitted. https://arxiv.org/abs/2506.20934
  60. T. M. Dunster, Asymptotic expansions for Legendre functions via differential equations having coalescing turning points. Stud. Appl. Math. (2025), Submitted. https://arxiv.org/abs/2504.19405
  61. T. M. Dunster, Uniform asymptotic expansions for generalised trigonometric integrals and their zeros. J. Math. Anal. Appl. (2025), Submitted. https://arxiv.org/abs/2508.05872


Book Chapter


             Legendre and related functions. NIST handbook of mathematical functions, 351-381, U.S. Dept. Commerce, Washington, DC, 2010, http://dlmf.nist.gov/14