purely imaginary
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1: 22.17 Moduli Outside the Interval [0,1]
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§22.17(i) Real or Purely Imaginary Moduli
►Jacobian elliptic functions with real moduli in the intervals and , or with purely imaginary moduli are related to functions with moduli in the interval by the following formulas. …2: 28.9 Zeros
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►Furthermore, for
and also have purely imaginary zeros that correspond uniquely to the purely imaginary
-zeros of (§10.21(i)), and they are asymptotically equal as and .
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3: 14.34 Software
4: Bibliography Z
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Mathieu functions for purely imaginary parameters.
J. Comput. Appl. Math. 236 (17), pp. 4513–4524.
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5: Bibliography I
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Bounds for the small real and purely imaginary zeros of Bessel and related functions.
Methods Appl. Anal. 2 (1), pp. 1–21.
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6: 10.30 Limiting Forms
7: Bibliography D
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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.
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Conical functions of purely imaginary order and argument.
Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.
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8: 20.2 Definitions and Periodic Properties
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9: 22.19 Physical Applications
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►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)).
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