of infinity
(0.003 seconds)
11—20 of 488 matching pages
11: 1.17 Integral and Series Representations of the Dirac Delta
…
►for all functions that are continuous when , and for each , converges absolutely for all sufficiently large values of .
The last condition is satisfied, for example, when as , where is a real constant.
►More generally, assume is piecewise continuous (§1.4(ii)) when for any finite positive real value of , and for each , converges absolutely for all sufficiently large values of .
…
►provided that is continuous when , and for each , converges absolutely for all sufficiently large values of (as in the case of (1.17.6)).
…
►The sum does not converge, but (1.17.18) can be interpreted as a generalized integral in the sense that
…
12: 2.2 Transcendental Equations
…
►Let be continuous and strictly increasing when and
►
2.2.1
.
►Then for the equation has a unique root in , and
►
2.2.2
.
…
►
2.2.4
.
…
13: 10.53 Power Series
14: 12.20 Approximations
…
►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions and (§13.2(i)) whose regions of validity include intervals with endpoints and , respectively.
As special cases of these results a Chebyshev-series expansion for valid when follows from (12.7.14), and Chebyshev-series expansions for and valid when follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13).
…
15: 20.8 Watson’s Expansions
…
►
20.8.1
…
16: 27.4 Euler Products and Dirichlet Series
17: 18.24 Hahn Class: Asymptotic Approximations
…
►With and fixed, Qiu and Wong (2004) gives an asymptotic expansion for as , that holds uniformly for .
…
►For two asymptotic expansions of as , with and fixed, see Jin and Wong (1998) and Wang and Wong (2011).
…
►Dunster (2001b) provides various asymptotic expansions for as , in terms of elementary functions or in terms of Bessel functions.
Taken together, these expansions are uniformly valid for and for in unbounded intervals—each of which contains , where again denotes an arbitrary small positive constant.
…
►For an asymptotic expansion of as , with fixed, see Li and Wong (2001).
…
18: 1.15 Summability Methods
…
►As
…
►If is integrable on , then
…
►Suppose now is real-valued and integrable on .
…where and .
…
►If is integrable on , then
…
19: 6.14 Integrals
20: 9.19 Approximations
…
►
•
►
•
…
►These expansions are for real arguments and are supplied in sets of four for each function, corresponding to intervals , , , .
…
►
•
Martín et al. (1992) provides two simple formulas for approximating to graphical accuracy, one for , the other for .
Moshier (1989, §6.14) provides minimax rational approximations for calculating , , , . They are in terms of the variable , where when is positive, when is negative, and when . The approximations apply when , that is, when or . The precision in the coefficients is 21S.
MacLeod (1994) supplies Chebyshev-series expansions to cover for and for . The Chebyshev coefficients are given to 20D.