Bessel functions and Hankel functions
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11—20 of 66 matching pages
11: 9.6 Relations to Other Functions
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§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions
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9.6.6
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9.6.8
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§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions
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9.6.20
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12: 10.3 Graphics
§10.3 Graphics
►§10.3(i) Real Order and Variable
►For the modulus and phase functions , , , and see §10.18. … ►§10.3(ii) Real Order, Complex Variable
… ►§10.3(iii) Imaginary Order, Real Variable
…13: 10.19 Asymptotic Expansions for Large Order
§10.19 Asymptotic Expansions for Large Order
►§10.19(i) Asymptotic Forms
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10.19.2
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§10.19(iii) Transition Region
… ►See also §10.20(i).14: 10.73 Physical Applications
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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)):
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15: 10.17 Asymptotic Expansions for Large Argument
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§10.17(i) Hankel’s Expansions
… ► … ►§10.17(iii) Error Bounds for Real Argument and Order
… ►§10.17(v) Exponentially-Improved Expansions
… ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).16: 10.51 Recurrence Relations and Derivatives
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►Let denote any of , , , or .
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17: 10.23 Sums
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§10.23(i) Multiplication Theorem
… ►§10.23(ii) Addition Theorems
… ► … ►For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).18: 10.27 Connection Formulas
19: 10.20 Uniform Asymptotic Expansions for Large Order
§10.20 Uniform Asymptotic Expansions for Large Order
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10.20.6
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10.20.9
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