integrals of modified Bessel functions
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31—40 of 61 matching pages
31: 7.6 Series Expansions
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§7.6(i) Power Series
… ►§7.6(ii) Expansions in Series of Spherical Bessel Functions
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7.6.8
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7.6.9
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7.6.10
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32: 9.17 Methods of Computation
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§9.17(iii) Integral Representations
►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing in the complex plane, once values of this function can be generated on the positive real axis. … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). … ►§9.17(iv) Via Bessel Functions
►In consequence of §9.6(i), algorithms for generating Bessel functions, Hankel functions, and modified Bessel functions (§10.74) can also be applied to , , and their derivatives. …33: 8.7 Series Expansions
§8.7 Series Expansions
►For the functions , , and see (8.4.11), §§10.47(ii), and 18.3, respectively. … ►
8.7.4
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8.7.5
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►For an expansion for in series of Bessel functions
that converges rapidly when and () is small or moderate in magnitude see Barakat (1961).
34: Bibliography L
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Bounds for modified Bessel functions.
J. Comput. Appl. Math. 34 (3), pp. 263–267.
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Numerical evaluation of integrals containing a spherical Bessel function by product integration.
J. Math. Phys. 22 (7), pp. 1399–1413.
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An application of the finite element approximation method to find the complex zeros of the modified Bessel function
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Math. Comp. 33 (148), pp. 1299–1306.
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Evaluation of Bessel function integrals with algebraic singularities.
J. Comput. Appl. Math. 37 (1-3), pp. 101–112.
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Integrals of Bessel Functions.
McGraw-Hill Book Co., Inc., New York.
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35: 11.13 Methods of Computation
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§11.13(i) Introduction
… ►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of , they are cumbersome to use when is large owing to slowness of convergence and cancellation. … ►Subsequently and are obtainable via (11.2.5) and (11.2.6). Other integrals that appear in §11.5(i) have highly oscillatory integrands unless is small. … ►Sequences of values of and , with fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). …36: 2.8 Differential Equations with a Parameter
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►Solutions are Bessel functions, or modified Bessel functions, of order (§§10.2, 10.25).
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►For and see §10.25(ii).
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►Define
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►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv).
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►For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii).
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37: Bibliography S
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A code to evaluate modified Bessel functions based on the continued fraction method.
Comput. Phys. Comm. 105 (2-3), pp. 263–272.
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Bounds for ratios of modified Bessel functions and associated Turán-type inequalities.
J. Math. Anal. Appl. 374 (2), pp. 516–528.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Computation of infinite integrals involving Bessel functions of arbitrary order by the -transformation.
J. Comput. Appl. Math. 78 (1), pp. 125–130.
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On the calculation of complex zeros of the modified Bessel function of the second kind.
Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.
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38: 10.60 Sums
§10.60 Sums
►§10.60(i) Addition Theorems
… ►§10.60(ii) Duplication Formulas
… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). ►§10.60(iv) Compendia
…39: Bibliography B
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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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Integral representations and summations of the modified Struve function.
Acta Math. Hungar. 141 (3), pp. 254–281.
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On numerical evaluation of integrals involving Bessel functions.
Apl. Mat. 31 (5), pp. 396–410.
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On the derivatives of the Bessel and Struve functions with respect to the order.
Integral Transforms Spec. Funct. 16 (3), pp. 187–198.
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Algorithm 484: Evaluation of the modified Bessel functions K0(Z) and K1(Z) for complex arguments.
Comm. ACM 17 (9), pp. 524–526.
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40: Bibliography R
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Tablitsy modifitsirovannykh funktsii Besselya
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“Nauka”, Moscow (Russian).
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Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions.
ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
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Applied Bessel Functions.
Dover Publications Inc., New York.
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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral.
Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.
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Partial fractions expansions and identities for products of Bessel functions.
J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
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