of%20imaginary%20argument
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1—10 of 12 matching pages
1: 5.22 Tables
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►Abramowitz and Stegun (1964, Chapter 6) tabulates , , , and for to 10D; and for to 10D; , , , , , , , and for to 8–11S; for to 20S.
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►Abramov (1960) tabulates for () , () to 6D.
Abramowitz and Stegun (1964, Chapter 6) tabulates for () , () to 12D.
This reference also includes for the same arguments to 5D.
Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of , , and for , to 8S.
2: Software Index
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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10.77(vi) Bessel Functions–Imaginary Order and Real Argument | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||||
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16.27(ii) Real Arguments | ✓ | ✓ | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||
16.27(iii) Complex Arguments | ✓ | a | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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22.22(ii) Real Argument | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |||||||||
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3: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials.
Math. Comp. 15 (73), pp. 7–11.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Generalized hypergeometric functions at unit argument.
Proc. Amer. Math. Soc. 114 (1), pp. 145–153.
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4: Bibliography D
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Bessel functions and of integer order and complex argument.
Comput. Phys. Comm. 78 (1-2), pp. 181–189.
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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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5: 20.10 Integrals
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20.10.1
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20.10.2
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20.10.3
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►Let , , and be constants such that , , and .
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►For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193).
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6: 9.18 Tables
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Miller (1946) tabulates , for , for ; , for ; , for ; , , , (respectively , , , ) for . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.
Zhang and Jin (1996, p. 337) tabulates , , , for to 8S and for to 9D.
Sherry (1959) tabulates , , , , ; 20S.
Corless et al. (1992) gives the real and imaginary parts of for ; 14S.
7: Bibliography G
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Algorithm 259: Legendre functions for arguments larger than one.
Comm. ACM 8 (8), pp. 488–492.
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Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities.
J. Math. Phys. 25 (11), pp. 3350–3356.
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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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8: 22.3 Graphics
9: 20.11 Generalizations and Analogs
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20.11.1
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►If both are positive, then allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):
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20.11.2
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►With the substitutions , , with , we have
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