Hankel%20inversion%20theorem
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1: 10.1 Special Notation
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►The main functions treated in this chapter are the Bessel functions , ; Hankel functions , ; modified Bessel functions , ; spherical Bessel functions , , , ; modified spherical Bessel functions , , ; Kelvin functions , , , .
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►Abramowitz and Stegun (1964): , , , , for , , , , respectively, when .
►Jeffreys and Jeffreys (1956): for , for , for .
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►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
2: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
… ►Inverse Hyperbolic Sine
… ►Inverse Hyperbolic Cosine
… ►Inverse Hyperbolic Tangent
… ►Other Inverse Functions
…3: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
… ►Inverse Sine
… ►Inverse Cosine
… ►Inverse Tangent
… ►Other Inverse Functions
…4: 22.15 Inverse Functions
§22.15 Inverse Functions
►§22.15(i) Definitions
… ►Each of these inverse functions is multivalued. The principal values satisfy … ►5: 10.3 Graphics
6: 10.75 Tables
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Achenbach (1986) tabulates , , , , , 20D or 18–20S.
§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives
… ►Döring (1966) tabulates all zeros of , , , , that lie in the sector , , to 10D. Some of the smaller zeros of and for are also included.
Kerimov and Skorokhodov (1985b) tabulates 50 zeros of the principal branches of and , 8D.
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
7: 10.73 Physical Applications
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►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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►More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging.
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►The functions , , , and arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates (§1.5(ii)):
…With the spherical harmonic defined as in §14.30(i), the solutions are of the form with , , , or , depending on the boundary conditions.
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8: Bibliography R
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Fourier analysis and signal processing by use of the Möbius inversion formula.
IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
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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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Finite-sum rules for Macdonald’s functions and Hankel’s symbols.
Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
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9: 20 Theta Functions
Chapter 20 Theta Functions
…10: Bibliography C
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Determination of -zeros of Hankel functions.
Comput. Phys. Comm. 32 (3), pp. 333–339.
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A quadrature formula for the Hankel transform.
Numer. Algorithms 9 (2), pp. 343–354.
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Numerical integration of related Hankel transforms by quadrature and continued fraction expansion.
Geophysics 48 (12), pp. 1671–1686.
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Optimized fast Hankel transform filters.
Geophysical Prospecting 38 (5), pp. 545–568.
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Computation of Hankel transforms.
SIAM Rev. 14 (2), pp. 278–285.
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