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 MÎş,iÎĽ(z)M_{\kappa,i\mu}(z) and WÎş,iÎĽ(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 MÎş,ÎĽ(z)M_{\kappa,\mu}(z) and WÎş,ÎĽ(z)W_{\kappa,\mu}(z) with ÎĽ\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-AntolĂ­n 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/S021953052550013
  55. T. M. Dunster, A. Gil, D. Ruiz-AntolĂ­n 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
  56. T. M. Dunster, A. Gil, D. Ruiz-AntolĂ­n 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
  57. T. M. Dunster, Asymptotic expansions for solutions of differential equations having coalescing turning points, with an application to Legendre functions. Stud. Appl. Math. 155 (5) (2025), p. e70138, https://doi.org/10.1111/sapm.70138
  58. T. M. Dunster, Simplified uniform asymptotic expansions for associated Legendre and conical functions. J. Approx. Theory 313 (2026), Article 106228, https://doi.org/10.1016/j.jat.2025.106228
  59. T. M. Dunster, Uniform asymptotic expansions for generalised trigonometric integrals and their zeros. J. Math. Anal. Appl. 560 (1) (2026), 130493, https://doi.org/10.1016/j.jmaa.2026.130493
  60. T. M. Dunster, Simplified Airy function asymptotic expansions for reverse generalised Bessel polynomials. J. Classical Anal. 28 (1) (2026), https://doi.org/10.7153/jca-2026-28-01
  61. T. M. Dunster, A. Gil, D. Ruiz-AntolĂ­n and J. Segura, Uniform asymptotic approximation and numerical evaluation of the reverse generalized Bessel polynomial zeros. Electron. Trans. Numer. Anal. 65 (2026), 140-155, https://doi.org/10.1553/etna_vol65s140
  62. T. M. Dunster, Asymptotic expansions for solutions of differential equations having a coalescing turning point and double pole, with an application to Legendre functions. Stud. Appl. Math. (2026), p. e70240, https://doi.org/10.1111/sapm.70240
  63. T. M. Dunster, Uniform asymptotic expansions for Bessel functions of imaginary order and their zeros. Integral Transforms Spec. Funct. 37 (7) (2026), 497-528, https://doi.org/10.1080/10652469.2025.2524721
  64. T. M. Dunster, The general Brannan coefficient conjecture and Watson's lemma. Anal. Appl. (2026), Submitted. https://arxiv.org/abs/2602.15308
  65. T. M. Dunster, The general Brannan coefficient conjecture II: Meijer-function approximations. Anal. Appl. (2026), Submitted. https://arxiv.org/abs/2606.11621

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