comparison with Clenshaw–Curtis formula

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21: 1.17 Integral and Series Representations of the Dirac Delta

… ►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): …Hence comparison with (1.17.5) shows that (1.17.9) can be interpreted as a generalized integral (1.17.3) with …Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation … ►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …

22: Bibliography O

… ►

F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §1.8(v), §22.11, §4.27.
Encodings:: BibTeX

… ►

J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

ⓘ

External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1967b) Bounds for the solutions of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.

ⓘ

External Links:: MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: §2.9(i).
Encodings:: BibTeX

… ►

F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

ⓘ

External Links:: Document, MathReview (Anthony Leung), ZentralBlatt
Referenced by:: §2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).
Encodings:: BibTeX

►

F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

ⓘ

External Links:: Document, MathReview (W. Wasow), ZentralBlatt
Referenced by:: §2.8(v).
Encodings:: BibTeX

…

23: 2.4 Contour Integrals

… ►For large

t

, the asymptotic expansion of

q(t)

may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function

F(z)

for

Q(z)

that has an inverse transform ►

2.4.6

f(t)=\frac{1}{2\pi i}\lim\limits_{\eta\to\infty}\int_{\sigma-i\eta}^{\sigma+i% \eta}e^{tz}F(z)\,\mathrm{d}z

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sigma$ : constant, $F(z)$ : comparison function and $f(x)$ : inverse transform of comparison function
Permalink:: http://dlmf.nist.gov/2.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §2.4(ii), §2.4 and Ch.2

… ►

2.4.7

q(t)-f(t)=\frac{e^{\sigma t}}{2\pi}\lim\limits_{\eta\to\infty}\int_{-\eta}^{% \eta}e^{it\tau}(Q(\sigma+i\tau)-F(\sigma+i\tau))\,\mathrm{d}\tau.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $q(t)$ : analytic function, $Q(z)$ : Laplace transform of $q(t)$ , $\sigma$ : constant, $F(z)$ : comparison function and $f(x)$ : inverse transform of comparison function
Permalink:: http://dlmf.nist.gov/2.4.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §2.4(ii), §2.4 and Ch.2

… ►

2.4.8

q(t)=f(t)+o\left(e^{ct}\right),

t\to+\infty

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $o\left(\NVar{x}\right)$ : order less than, $q(t)$ : analytic function, $c$ : real constant and $f(x)$ : inverse transform of comparison function
Permalink:: http://dlmf.nist.gov/2.4.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §2.4(ii), §2.4 and Ch.2

… ►

2.4.9

q(t)=f(t)+o\left(t^{-m}e^{ct}\right),

t\to+\infty

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $o\left(\NVar{x}\right)$ : order less than, $q(t)$ : analytic function, $c$ : real constant and $f(x)$ : inverse transform of comparison function
Permalink:: http://dlmf.nist.gov/2.4.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §2.4(ii), §2.4 and Ch.2

…

24: 22.21 Tables

… ►Curtis (1964b) tabulates

\operatorname{sn}\left(mK/n,k\right)

\operatorname{cn}\left(mK/n,k\right)

\operatorname{dn}\left(mK/n,k\right)

for

n=2(1)15

m=1(1)n-1

, and

q

(not

k

)

=0(.005)0.35

to 20D. … ►Tables of theta functions (§20.15) can also be used to compute the twelve Jacobian elliptic functions by application of the quotient formulas given in §22.2.

25: 3.3 Interpolation

… ►

Three-Point Formula

… ►

Four-Point Formula

… ►

Five-Point Formula

… ►

Six-Point Formula

… ►For comparison, we use Newton’s interpolation formula (3.3.38) …

26: Bibliography B

… ►

R. W. Barnard, K. Pearce, and K. C. Richards (2000) A monotonicity property involving

{}_{3}F_{2}

and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.

ⓘ

External Links:: ISSN 1095-7154, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §19.9(i), §19.9(i).
Encodings:: BibTeX

… ►

W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

ⓘ

External Links:: ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

ⓘ

External Links:: Document, MathReview (T. M. Apostol), ZentralBlatt
Referenced by:: §24.16(ii).
Encodings:: BibTeX

►

B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

ⓘ

External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.16(iii), §24.8(ii).
Encodings:: BibTeX

… ►

R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with

\exp(-x^{4})

. J. Approx. Theory 98, pp. 146–166.

ⓘ

External Links:: ISSN 0021-9045, Document, MathReview (V. L. Deshpande), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

…

27: Bibliography S

… ►

B. Simon (2005c) Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.

ⓘ

External Links:: Document, Link, MathReview Entry
Referenced by:: §18.36(vi), §18.39(i).
Encodings:: BibTeX

… ►

R. Spira (1971) Calculation of the gamma function by Stirling’s formula. Math. Comp. 25 (114), pp. 317–322.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §5.11(i), §5.11(ii).
Encodings:: BibTeX

… ►

A. H. Stroud and D. Secrest (1966) Gaussian Quadrature Formulas. Prentice-Hall Inc., Englewood Cliffs, N.J..

ⓘ

External Links:: MathReview (P. C. Hammer), ZentralBlatt
Referenced by:: §3.5(v).
Encodings:: BibTeX

…

28: 4.47 Approximations

… ►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for

\ln

\exp

\sin

\cos

\tan

\cot

\operatorname{arcsin}

\operatorname{arctan}

\operatorname{arcsinh}

. …

29: 2.11 Remainder Terms; Stokes Phenomenon

… ►

§2.11(ii) Connection Formulas

… ►However, on combining (2.11.6) with the connection formula (8.19.18), with

m=1

, we derive … ►Comparison with the true value …

30: 4.23 Inverse Trigonometric Functions

… ►

§4.23(iii) Reflection Formulas

…