# asymptotic approximations of sums and sequences

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##### 2: 2.1 Definitions and Elementary Properties
means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left((x-c)^{n}\right)$ as $x\to c$ in $\mathbf{X}$. …
###### §2.1(v) Generalized Asymptotic Expansions
Then $\{\phi_{s}(x)\}$ is an asymptotic sequence or scale. Suppose also that $f(x)$ and $f_{s}(x)$ satisfy …
##### 3: 5.4 Special Values and Extrema
5.4.7 $\Gamma\left(\tfrac{1}{3}\right)=2.67893\;85347\;07747\;63365\;\dots,$
5.4.8 $\Gamma\left(\tfrac{2}{3}\right)=1.35411\;79394\;26400\;41694\;\dots,$
5.4.9 $\Gamma\left(\tfrac{1}{4}\right)=3.62560\;99082\;21908\;31193\;\dots,$
As $n\to\infty$,
##### 4: Bibliography W
• J. K. G. Watson (1999) Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.
• E. J. Weniger (1996) Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.
• E. J. Weniger (2007) Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.
• R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.
• R. Wong (1995) Error bounds for asymptotic approximations of special functions. Ann. Numer. Math. 2 (1-4), pp. 181–197.
• ##### 5: 4.45 Methods of Computation
Beginning with $x_{0}=x$, generate the sequenceAnother method, when $x$ is large, is to sumInitial approximations are obtainable, for example, from the power series (4.13.6) (with $t\geq 0$) when $x$ is close to $-1/e$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …
##### 6: Bibliography R
• A. Ralston (1965) Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.
• M. Razaz and J. L. Schonfelder (1981) Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.
• W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.
• W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.
• RISC Combinatorics Group (website) Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.
• ##### 7: Bibliography S
• C. W. Schelin (1983) Calculator function approximation. Amer. Math. Monthly 90 (5), pp. 317–325.
• M. J. Seaton (1984) The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.
• D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.
• N. J. A. Sloane (2003) The On-Line Encyclopedia of Integer Sequences. Notices Amer. Math. Soc. 50 (8), pp. 912–915.
• A. H. Stroud (1971) Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, N.J..
• ##### 8: Bibliography L
• H. A. Lauwerier (1974) Asymptotic Analysis. Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam.
• L. Lorch and M. E. Muldoon (2008) Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.
• Y. L. Luke (1968) Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.
• Y. L. Luke (1970) Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.
• Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.
• ##### 9: Bibliography C
• B. C. Carlson and J. L. Gustafson (1994) Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25 (2), pp. 288–303.
• E. W. Cheney (1982) Introduction to Approximation Theory. 2nd edition, Chelsea Publishing Co., New York.
• W. J. Cody (1968) Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.
• W. J. Cody (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.
• Combinatorial Object Server (website) Department of Computer Science, University of Victoria, Canada.
• ##### 10: 25.11 Hurwitz Zeta Function
where …
###### §25.11(xi) Sums
For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360).
###### §25.11(xii) $a$-Asymptotic Behavior
As $a\to\infty$ in the sector $|\operatorname{ph}a|\leq\pi-\delta(<\pi)$, with $s(\neq 1)$ and $\delta$ fixed, we have the asymptotic expansion …