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1: 2.11 Remainder Terms; Stokes Phenomenon

… ►When a rigorous bound or reliable estimate for the remainder term is unavailable, it is unsafe to judge the accuracy of an asymptotic expansion merely from the numerical rate of decrease of the terms at the point of truncation. … ►If the results agree within

S

significant figures, then it is likely—but not certain—that the truncated asymptotic series will yield at least

S

correct significant figures for larger values of

x

. … ►Truncation after 5 terms yields 0. … ►The process just used is equivalent to re-expanding the remainder term of the original asymptotic series (2.11.24) in powers of

1/(x+5)

and truncating the new series optimally. … ►Optimum truncation occurs just prior to the numerically smallest term, that is, at

s_{4}

. …

2: 8.11 Asymptotic Approximations and Expansions

… ►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. …

3: 2.5 Mellin Transform Methods

… ►First, we introduce the truncated functions

f_{1}(t)

and

f_{2}(t)

defined by ►

2.5.21

f_{1}(t)=\begin{cases}f(t),&0<t\leq 1,\\ 0,&1<t<\infty,\end{cases}

ⓘ

Symbols:: $f(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Permalink:: http://dlmf.nist.gov/2.5.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §2.5(ii), §2.5 and Ch.2

►

2.5.22

f_{2}(t)=f(t)-f_{1}(t).

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Symbols:: $f(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Permalink:: http://dlmf.nist.gov/2.5.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §2.5(ii), §2.5 and Ch.2

… ►

2.5.26

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muf_{1% }\mskip 3.0mu\left(z\right)+\mathscr{M}\mskip-3.0muf_{2}\mskip 3.0mu\left(z\right)

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Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $f(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Keywords:: Mellin transform
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E26
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(ii), §2.5 and Ch.2

… ►

2.5.30

I_{jk}(x)=\int_{0}^{\infty}f_{j}(t)h_{k}(xt)\,\mathrm{d}t.

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $h(x)$ : locally integrable function, $f_{j}(t)$ : truncated functions and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §2.5(ii), §2.5 and Ch.2

…

4: 3.11 Approximation Techniques

… ►Because the series (3.11.12) converges rapidly we obtain a very good first approximation to the minimax polynomial

p_{n}(x)

for

[a,b]

if we truncate (3.11.12) at its

(n+1)

th term. … ►Since

L_{0}=1

L_{n}

is a monotonically increasing function of

n

, and (for example)

L_{1000}=4.07\dots

, this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. … ►Let

c_{n}T_{n}\left(x\right)

be the last term retained in the truncated series. …Then the sum of the truncated expansion equals

\frac{1}{2}(u_{0}-u_{2})

. …

5: 12.11 Zeros

…

6: 22.3 Graphics

… ►

See accompanying text — Figure 22.3.26: Density plot of $|\operatorname{sn}\left(5,k\right)|$ as a function of complex $k^{2}$ , $-10\leq\Re\left(k^{2}\right)\leq 20$ , $-10\leq\Im\left(k^{2}\right)\leq 10$ . Grayscale, running from 0 (black) to 10 (white), with $|(\operatorname{sn}\left(5,k\right))|>10$ truncated to 10. … Magnify

►

7: Bibliography W

… ►

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.

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