Taylor theorem

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1: 12.13 Sums

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§12.13(i) Addition Theorems

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2: 1.5 Calculus of Two or More Variables

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§1.5(iii) Taylor’s Theorem; Maxima and Minima

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3: 1.4 Calculus of One Variable

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§1.4(vi) Taylor’s Theorem for Real Variables

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4: 1.10 Functions of a Complex Variable

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§1.10(i) Taylor’s Theorem for Complex Variables

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5: 25.11 Hurwitz Zeta Function

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25.11.10 $$\zeta (s,a)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_n}{n!}\zeta \left(n+s\right){(1-a)}^n,$$ $s\ne 1$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

infinite series, series representation

Proof sketch:

Derivable using Taylor’s theorem (1.10.1) with $z=a$, $z_0=1$, (25.11.17), (5.2.4), (25.11.2).

Referenced by:

§25.11(iv), §25.11(iv), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.11.E10

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n+s\right)}{n!\mathrm{\Gamma }\left(s\right)}\zeta \left(n+s\right){(1-a)}^n$ more concisely as ${\sum }_{n=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_n}{n!}\zeta \left(n+s\right){(1-a)}^n$ using the Pochhammer symbol.

See also:

Annotations for §25.11(iv), §25.11 and Ch.25

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6: Bibliography D

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Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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The Taylor Series. Oxford University Press, Oxford.

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Notes:

Reprinted by Dover Publications Inc., New York, 1957

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.9(iii).

Encodings:

BibTeX

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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External Links:

ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

Encodings:

BibTeX

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

BibTeX

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7: Bibliography M

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Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edition, Springer-Verlag, New York-Berlin.

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Notes:

Table errata: Math. Comp. v. 23 (1969), p. 471, v. 24 (1970), pp. 240 and 505, v. 25 (1971), no. 113, p. 201, v. 29 (1975), no. 130, p. 672, v. 30 (1976), no. 135, pp. 677–678, v. 32 (1978), no. 141, pp. 319–320, v. 36 (1981), no. 153, pp. 315–317, v. 41 (1983), no. 164, pp. 775–776, and v. 81 (2012), no. 280, p. 2251

External Links:

MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§10.15, §10.22(vi), §10.32(iv), §10.38, §10.43(vi), §10.60(iv), §10.9(iv), §11.5(iii), Ch.11, §12.12, §12.5(iv), Ch.12, §13.10(vi), §13.23(iv), §14.1, §14.11, §14.12(ii), §14.18(iv), §14.25, §14.3(iii), §14.5(v), §14.7(iv), Ch.14, (25.5.21), §25.5, (25.6.1), (25.6.2), §25.6(i), §8.1, Notations B, Notations I, Notations P, Notations P, Notations Q, Notations Q.

Encodings:

BibTeX

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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External Links:

ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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CODATA recommended values of the fundamental physical constants: 2002. Rev. Mod.Phys. 77, pp. 1–107.

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Referenced by:

§18.39(ii).

Encodings:

BibTeX

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8: Bibliography B

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1897 original, with a foreword by Igor Krichever.

External Links:

ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt

Referenced by:

§20.9(ii).

Encodings:

BibTeX

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A general program to calculate atomic continuum processes using the R-matrix method. Comput. Phys. Comm. 8 (3), pp. 149–198.

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External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

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Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.

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External Links:

Document

Referenced by:

§36.14(i).

Encodings:

BibTeX

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9: Bibliography K

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An extension of Saalschütz’s summation theorem for the series ${}_{r+3}{}^{}F_{r+2}^{}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt

Referenced by:

§16.4(ii), §16.4(ii).

Encodings:

BibTeX

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A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.10.

Encodings:

BibTeX

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

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Notes:

Translated from the Russian by E. V. Pankratiev

External Links:

ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt

Referenced by:

§10.73(i), §10.73(i), §10.73(i).

Encodings:

BibTeX

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10: 2.10 Sums and Sequences

… ►The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. … ►and Cauchy’s theorem, we have … ►

§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

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2.10.25 $$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{z}^n,$$ .

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Symbols:

$f(x)$: analytic function, $f_n$: coefficients and $r$: radious of annulus

Permalink:

http://dlmf.nist.gov/2.10.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

►What is the asymptotic behavior of $f_n$ as $n\to \mathrm{\infty }$ or $n\to -\mathrm{\infty }$? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? …