of%20imaginary%20order
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1—10 of 15 matching pages
1: 10.75 Tables
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Zhang and Jin (1996, pp. 185–195) tabulates , , , , , , 5, 10, 25, 50, 100, 9S; , , , , , , , 8S; real and imaginary parts of , , , , , , , , 8S.
MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function , 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).
Zhang and Jin (1996, pp. 240–250) tabulates , , , , , , 9S; , , , , , 10, 30, 50, 100, , , , , , , 5, 10, 50, 8S; real and imaginary parts of , , , , , 20(10)50, 100, , , 8S.
§10.75(viii) Modified Bessel Functions of Imaginary or Complex Order
… ►Žurina and Karmazina (1967) tabulates for , , 7S.
2: 10.3 Graphics
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§10.3(i) Real Order and Variable
… ►§10.3(ii) Real Order, Complex Variable
… ►§10.3(iii) Imaginary Order, Real Variable
… ► ►3: Bibliography D
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A simplified algorithm for the second-order sound fields.
J. Acoust. Soc. Amer. 108 (6), pp. 2759–2764.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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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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4: 18.40 Methods of Computation
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►It is now necessary to take the limit of , and the imaginary part is the required Stieltjes–Perron inversion:
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18.40.6
►The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
Results of low ( to decimal digits) precision for are easily obtained for to .
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5: 11.6 Asymptotic Expansions
6: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Asymptotic expansions of the modified Bessel function of the third kind of imaginary order.
SIAM J. Appl. Math. 15, pp. 1315–1323.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Orthogonality relations for the associated Legendre functions of imaginary order.
Integral Transforms Spec. Funct. 24 (4), pp. 331–337.
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Bounds and algorithms for the -Bessel function of imaginary order.
LMS J. Comput. Math. 16, pp. 78–108.
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7: Software Index
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8: 36.5 Stokes Sets
9: 12.11 Zeros
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12.11.1
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12.11.2
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12.11.3
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►For example, let the th real zeros of and , counted in descending order away from the point , be denoted by and , respectively.
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12.11.9
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10: Bibliography G
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Evaluation of the modified Bessel function of the third kind of imaginary orders.
J. Comput. Phys. 175 (2), pp. 398–411.
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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.
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Algorithm 831: Modified Bessel functions of imaginary order and positive argument.
ACM Trans. Math. Software 30 (2), pp. 159–164.
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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments.
ACM Trans. Math. Software 30 (2), pp. 145–158.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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