Chinese remainder theorem

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11: 18.30 Associated OP’s

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Markov’s Theorem

►The ratio $p_n^{(0)}(z)/p_n(z)$, as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials. … ►

18.30.25 $$\underset{n\to \mathrm{\infty }}{lim}{F}_n(x)=\underset{n\to \mathrm{\infty }}{lim}{p}_n^{(0)}(z)/p_n(z)=\frac{1}{{\mu }_0}{\int }_a^b\frac{d\mu (x)}{z-x},$$ $z\in ℂ\backslash [a,b]$.

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Symbols:

$[a,b]$: closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Chihara (1978, Chapter III, Theorem 4.3)(proved)

Referenced by:

§18.30(vi), §18.30(vii)

Permalink:

http://dlmf.nist.gov/18.30.E25

Encodings:

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See also:

Annotations for §18.30(vi), §18.30(vi), §18.30 and Ch.18

… ►The ratio ${\widehat{p}}_{n-1}(x;1)/{\widehat{p}}_n(x)$ is then the $F_n(x)$ of (18.2.35), leading to Markov’s theorem as stated in (18.30.25). …

12: 6.12 Asymptotic Expansions

… ►When $|\mathrm{ph}z|\le \frac{1}{2}\pi$ the remainder is bounded in magnitude by the first neglected term, and has the same sign when $\mathrm{ph}z=0$. …For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$. ►For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv), with $p=1$. …If the expansion is terminated at the $n$th term, then the remainder term is bounded by $1+\chi (n+1)$ times the next term. … ►The remainder terms are given by …

13: 1.5 Calculus of Two or More Variables

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Implicit Function Theorem

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

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1.5.18 $$f(a+\lambda ,b+\mu )=f+\left(\lambda \frac{\partial }{\partial x}+\mu \frac{\partial }{\partial y}\right)f+\mathrm{\cdots }+\frac{1}{n!}{\left(\lambda \frac{\partial }{\partial x}+\mu \frac{\partial }{\partial y}\right)}^{n}f+R_n,$$

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Symbols:

$!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer and $R_n$: remainder

Referenced by:

§1.5(iii)

Permalink:

http://dlmf.nist.gov/1.5.E18

Encodings:

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See also:

Annotations for §1.5(iii), §1.5 and Ch.1

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§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals

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14: 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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An improvement of the remainder term in the divisor problem. Mat. Zametki 6, pp. 545–554 (Russian).

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Notes:

In Russian. English translation: Math. Notes 6(1969), no. 5, pp. 784–791.

External Links:

Document, MathReview (A. I. Vinogradov), ZentralBlatt

Referenced by:

§27.11, Erratum (V1.1.8) for Section 27.11.

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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15: 9.8 Modulus and Phase

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9.8.22 $$\theta \left(x\right)\sim \frac{\pi }{4}+\frac{2}{3}{(-x)}^{3/2}\left(1+\frac{5}{32}\frac{1}{x^3}+\frac{1105}{6144}\frac{1}{x^6}+\frac{82825}{65536}\frac{1}{x^9}+\frac{\mathrm{12820\hspace{0.33em}31525}}{\mathrm{587\hspace{0.33em}20256}}\frac{1}{x^{12}}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).

Referenced by:

§9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E22

Encodings:

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See also:

Annotations for §9.8(iv), §9.8 and Ch.9

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9.8.23 $$\varphi \left(x\right)\sim -\frac{\pi }{4}+\frac{2}{3}{(-x)}^{3/2}\left(1-\frac{7}{32}\frac{1}{x^3}-\frac{1463}{6144}\frac{1}{x^6}-\frac{\mathrm{4\hspace{0.33em}95271}}{\mathrm{3\hspace{0.33em}27680}}\frac{1}{x^9}-\frac{\mathrm{2065\hspace{0.33em}30429}}{\mathrm{83\hspace{0.33em}88608}}\frac{1}{x^{12}}-\mathrm{\cdots }\right).$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\varphi \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).

Referenced by:

§9.8(iv), §9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E23

Encodings:

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See also:

Annotations for §9.8(iv), §9.8 and Ch.9

►The remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n\ge 0$ for (9.8.20), (9.8.22) and (9.8.23), and $n\ge 1$ for (9.8.21). …

16: 2.10 Sums and Sequences

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2.10.4 $$S(n)=\frac{1}{2}{n}^2\ln n-\frac{1}{4}{n}^2+\frac{1}{2}n\ln n+\frac{1}{12}\ln n+C+\sum _{s=2}^{m-1}\frac{(-B_{2s})}{2s(2s-1)(2s-2)}\frac{1}{n^{2s-2}}+R_m(n),$$

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Symbols:

$B_n$: Bernoulli numbers, $\ln z$: principal branch of logarithm function, $m$: integer, $n$: integer, $S(n)$: sum, $C$: constant and $R_m(n)$: remainder

Referenced by:

§2.10(i)

Permalink:

http://dlmf.nist.gov/2.10.E4

Encodings:

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See also:

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

►where $m$ ($\ge 2$) is arbitrary, $C$ is a constant, and … ►In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\ge \frac{1}{2}(\alpha +1)$. … ►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 …

17: 13.13 Addition and Multiplication Theorems

§13.13 Addition and Multiplication Theorems

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§13.13(i) Addition Theorems for $M(a,b,z)$

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§13.13(ii) Addition Theorems for $U(a,b,z)$

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13.13.12 $${\mathrm{e}}^y{\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!x^n}U(a-n,b-n,x),$$ .

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

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§13.13(iii) Multiplication Theorems for $M(a,b,z)$ and $U(a,b,z)$

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18: 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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Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.

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External Links:

ISSN 0020-2932, Document, MathReview, ZentralBlatt

Referenced by:

§2.6(iii).

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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19: 10.23 Sums

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§10.23(i) Multiplication Theorem

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§10.23(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

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20: 27.16 Cryptography

… ►Thus, $y\equiv x^r(modn)$ and . … ►By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. …