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1: 10.24 Functions of Imaginary Order
J ~ - ν ( x ) = J ~ ν ( x ) ,
and J ~ ν ( x ) , Y ~ ν ( x ) are linearly independent solutions of (10.24.1): … In consequence of (10.24.6), when x is large J ~ ν ( x ) and Y ~ ν ( x ) comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … For graphs of J ~ ν ( x ) and Y ~ ν ( x ) see §10.3(iii). For mathematical properties and applications of J ~ ν ( x ) and Y ~ ν ( x ) , including zeros and uniform asymptotic expansions for large ν , see Dunster (1990a). …
2: 10.45 Functions of Imaginary Order
and I ~ ν ( x ) , K ~ ν ( x ) are real and linearly independent solutions of (10.45.1): … The corresponding result for K ~ ν ( x ) is given by … For graphs of I ~ ν ( x ) and K ~ ν ( x ) see §10.26(iii). For properties of I ~ ν ( x ) and K ~ ν ( x ) , including uniform asymptotic expansions for large ν and zeros, see Dunster (1990a). …
3: 10.26 Graphics
§10.26(iii) Imaginary Order, Real Variable
See accompanying text
Figure 10.26.7: I ~ 1 / 2 ( x ) , K ~ 1 / 2 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.8: I ~ 1 ( x ) , K ~ 1 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.9: I ~ 5 ( x ) , K ~ 5 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.10: K ~ 5 ( x ) , 0.01 x 3 . Magnify
4: 10.3 Graphics
§10.3(iii) Imaginary Order, Real Variable
See accompanying text
Figure 10.3.17: J ~ 1 / 2 ( x ) , Y ~ 1 / 2 ( x ) , 0.01 x 10 . Magnify
See accompanying text
Figure 10.3.18: J ~ 1 ( x ) , Y ~ 1 ( x ) , 0.01 x 10 . Magnify
See accompanying text
Figure 10.3.19: J ~ 5 ( x ) , Y ~ 5 ( x ) , 0.01 x 10 . Magnify
5: 10.74 Methods of Computation
§10.74(viii) Functions of Imaginary Order
For the computation of the functions I ~ ν ( x ) and K ~ ν ( x ) defined by (10.45.2) see Temme (1994c) and Gil et al. (2002d, 2003a, 2004b).
6: 10.76 Approximations
Real Variable; Imaginary Order
7: 10.77 Software
§10.77(vi) Bessel Functions–Imaginary Order and Real Argument
8: 10.75 Tables
  • MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function J 0 ( z ) - i J 1 ( z ) , 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).

  • §10.75(viii) Modified Bessel Functions of Imaginary or Complex Order
  • Žurina and Karmazina (1967) tabulates K ~ ν ( x ) for ν = 0.01 ( .01 ) 10 , x = 0.1 ( .1 ) 10.2 , 7S.

  • 9: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2002d) Evaluation of the modified Bessel function of the third kind of imaginary orders. J. Comput. Phys. 175 (2), pp. 398–411.
  • A. Gil, J. Segura, and N. M. Temme (2003a) Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. J. Comput. Appl. Math. 153 (1-2), pp. 225–234.
  • A. Gil, J. Segura, and N. M. Temme (2004a) Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Software 30 (2), pp. 159–164.
  • A. Gil, J. Segura, and N. M. Temme (2004b) Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.
  • 10: 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.