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1: 22.17 Moduli Outside the Interval [0,1]
§22.17(i) Real or Purely Imaginary Moduli
Jacobian elliptic functions with real moduli in the intervals ( - , 0 ) and ( 1 , ) , or with purely imaginary moduli are related to functions with moduli in the interval [ 0 , 1 ] by the following formulas. …
2: 28.9 Zeros
Furthermore, for q > 0 ce m ( z , q ) and se m ( z , q ) also have purely imaginary zeros that correspond uniquely to the purely imaginary z -zeros of J m ( 2 q cos z ) 10.21(i)), and they are asymptotically equal as q 0 and | z | . …
3: 14.34 Software
  • Gil and Segura (1998). Integer parameters and purely imaginary arguments. Fortran.

  • 4: Bibliography Z
  • C. H. Ziener, M. Rückl, T. Kampf, W. R. Bauer, and H. P. Schlemmer (2012) Mathieu functions for purely imaginary parameters. J. Comput. Appl. Math. 236 (17), pp. 4513–4524.
  • 5: Bibliography I
  • M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.
  • 6: 10.30 Limiting Forms
    For K ν ( x ) , when ν is purely imaginary and x 0 + , see (10.45.2) and (10.45.7). …
    7: Bibliography D
  • T. M. Dunster (1990a) 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), pp. 995–1018.
  • T. M. Dunster (2013) Conical functions of purely imaginary order and argument. Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.
  • 8: 20.2 Definitions and Periodic Properties
    Figure 20.2.1: z -plane. …Left-hand diagram is the rectangular case ( τ purely imaginary); right-hand diagram is the general case. …
    9: 22.19 Physical Applications
    Its dynamics for purely imaginary time is connected to the theory of instantons (Itzykson and Zuber (1980, p. 572), Schäfer and Shuryak (1998)), to WKB theory, and to large-order perturbation theory (Bender and Wu (1973), Simon (1982)). …
    10: 33.22 Particle Scattering and Atomic and Molecular Spectra
    At negative energies E < 0 and both ρ and η are purely imaginary. …