expansion of arbitrary function
(0.006 seconds)
11—20 of 64 matching pages
11: 11.9 Lommel Functions
12: 33.12 Asymptotic Expansions for Large
…
►The first set is in terms of Airy functions and the expansions are uniform for fixed and , where is an arbitrary small positive constant.
…
13: 12.9 Asymptotic Expansions for Large Variable
14: 13.7 Asymptotic Expansions for Large Argument
§13.7 Asymptotic Expansions for Large Argument
… ►§13.7(ii) Error Bounds
… ►§13.7(iii) Exponentially-Improved Expansion
… ►where is an arbitrary nonnegative integer, and … ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).15: 9.12 Scorer Functions
§9.12 Scorer Functions
… ►where and are arbitrary constants, and are any two linearly independent solutions of Airy’s equation (9.2.1), and is any particular solution of (9.12.1). … ►§9.12(viii) Asymptotic Expansions
… ►As , and with denoting an arbitrary small positive constant, … ►Integrals
…16: 12.14 The Function
§12.14 The Function
… ►§12.14(v) Power-Series Expansions
… ►§12.14(viii) Asymptotic Expansions for Large Variable
… ►In the following expansions, obtained from Olver (1959), is large and positive, and is again an arbitrary small positive constant. … ► …17: 14.20 Conical (or Mehler) Functions
…
►
…
►
§14.20(v) Trigonometric Expansion
… ►uniformly for , where and are the modified Bessel functions (§10.25(ii)) and is an arbitrary constant such that . For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). … ►In this subsection and §14.20(ix), and denote arbitrary constants such that and . …18: 13.19 Asymptotic Expansions for Large Argument
§13.19 Asymptotic Expansions for Large Argument
… ►Again, denotes an arbitrary small positive constant. … ►Error bounds and exponentially-improved expansions are derivable by combining §§13.7(ii) and 13.7(iii) with (13.14.2) and (13.14.3). … ►For an asymptotic expansion of as that is valid in the sector and where the real parameters , are subject to the growth conditions , , see Wong (1973a).19: 13.31 Approximations
§13.31 Approximations
►§13.31(i) Chebyshev-Series Expansions
►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of and that include the intervals and , respectively, where is an arbitrary positive constant. … ►
13.31.2
…
►
13.31.3