mean value theorem

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1: 1.4 Calculus of One Variable

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Mean Value Theorem

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First Mean Value Theorem

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Second Mean Value Theorem

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2: 1.6 Vectors and Vector-Valued Functions

… ►Note that $C$ can be given an orientation by means of $𝐜$. … ►

Green’s Theorem

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Stokes’s Theorem

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Gauss’s (or Divergence) Theorem

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Green’s Theorem (for Volume)

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3: Bille C. Carlson

… ►In theoretical physics he is known for the “Carlson-Keller Orthogonalization”, published in 1957, Orthogonalization Procedures and the Localization of Wannier Functions, and the “Carlson-Keller Theorem”, published in 1961, Eigenvalues of Density Matrices. … ►Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities. …

4: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Mean Value Property

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Dominated Convergence Theorem

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5: 1.10 Functions of a Complex Variable

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Picard’s Theorem

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§1.10(iv) Residue Theorem

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Rouché’s Theorem

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Lagrange Inversion Theorem

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Extended Inversion Theorem

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6: 2.1 Definitions and Elementary Properties

… ►(In other words $=$ here really means $\subseteq$.) … ►For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $𝐒$ and $f(z)=O\left(z^{\nu }\right)$ as $z\to \mathrm{\infty }$ in $𝐒$, $\nu$ being real, then $f^{\prime }(z)=O\left(z^{\nu -1}\right)$ as $z\to \mathrm{\infty }$ in any closed sector properly interior to $𝐒$ and with the same vertex (Ritt’s theorem). This result also holds with both $O$’s replaced by $o$’s. … ►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 $𝐗$. … ►As an example, in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ () each of the functions $0,{\mathrm{e}}^{-z}$, and ${\mathrm{e}}^{-\sqrt{z}}$ (principal value) has the null asymptotic expansion …

7: Bibliography F

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:

ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

BibTeX

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Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.

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External Links:

ISSN 0196-8858, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

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Notes:

Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp (i\theta )$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii), §19.37(ii).

Encodings:

BibTeX

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:

FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.

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External Links:

ISSN 0378-4371, Document, MathReview

Referenced by:

6th item.

Encodings:

BibTeX

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8: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … ►All cases of $R_F$, $R_C$, $R_J$, and $R_D$ are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). … ►This method loses significant figures in $\rho$ if ${\alpha }^2$ and $k^2$ are nearly equal unless they are given exact values—as they can be for tables. … ►When the values of complete integrals are known, addition theorems with $\psi =\pi /2$ (§19.11(ii)) ease the computation of functions such as $F(\varphi ,k)$ when $\frac{1}{2}\pi -\varphi$ is small and positive. …These special theorems are also useful for checking computer codes. …

9: 10.18 Modulus and Phase Functions

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10.18.17 $$M_{\nu }^2\left(x\right)\sim \frac{2}{\pi x}\left(1+\frac{1}{2}\frac{\mu -1}{{(2x)}^2}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu -1)(\mu -9)}{{(2x)}^4}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{(\mu -1)(\mu -9)(\mu -25)}{{(2x)}^6}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter

Proof sketch:

See Nemes (2021, Theorem 2.1)

A&S Ref:

9.2.28

Referenced by:

§10.18(iii), §10.18(iii), §10.40(i), §10.40(i), §10.49(iv), Erratum (V1.1.8) for Section 10.18(iii)

Permalink:

http://dlmf.nist.gov/10.18.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

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10.18.18 $${\theta }_{\nu }\left(x\right)\sim x-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi +\frac{\mu -1}{2(4x)}+\frac{(\mu -1)(\mu -25)}{6{(4x)}^3}+\frac{(\mu -1)({\mu }^2-114\mu +1073)}{5{(4x)}^5}+\frac{(\mu -1)(5{\mu }^3-1535{\mu }^2+54703\mu -\mathrm{3\hspace{0.33em}75733})}{14{(4x)}^7}+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

Proof sketch:

See Nemes (2021, Theorem 2.1)

A&S Ref:

9.2.29

Referenced by:

§10.18(iii), §10.18(iii), Erratum (V1.1.8) for Section 10.18(iii)

Permalink:

http://dlmf.nist.gov/10.18.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(iii), §10.18 and Ch.10

… ►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).