# of imaginary order

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##### 1: 10.24 Functions of Imaginary Order
$\widetilde{J}_{-\nu}\left(x\right)=\widetilde{J}_{\nu}\left(x\right),$
and $\widetilde{J}_{\nu}\left(x\right)$, $\widetilde{Y}_{\nu}\left(x\right)$ are linearly independent solutions of (10.24.1): … In consequence of (10.24.6), when $x$ is large $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … For graphs of $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ see §10.3(iii). For mathematical properties and applications of $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$, including zeros and uniform asymptotic expansions for large $\nu$, see Dunster (1990a). …
##### 2: 10.45 Functions of Imaginary Order
10.45.3 $\displaystyle\widetilde{I}_{-\nu}\left(x\right)=\widetilde{I}_{\nu}\left(x% \right),$ $\displaystyle\widetilde{K}_{-\nu}\left(x\right)=\widetilde{K}_{\nu}\left(x% \right),$
and $\widetilde{I}_{\nu}\left(x\right)$, $\widetilde{K}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1): … The corresponding result for $\widetilde{K}_{\nu}\left(x\right)$ is given by … For graphs of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ see §10.26(iii). For properties of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). …
##### 5: 10.74 Methods of Computation
###### §10.74(viii) Functions of ImaginaryOrder
For the computation of the functions $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ defined by (10.45.2) see Temme (1994c) and Gil et al. (2002d, 2003a, 2004b).
##### 8: 10.75 Tables
• MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function $J_{0}\left(z\right)-iJ_{1}\left(z\right)$, 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 $\widetilde{K}_{\nu}\left(x\right)$ for $\nu=0.01(.01)10$, $x=0.1(.1)10.2$, 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.