# mean value theorems

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## 6 matching pages

##### 1: 1.4 Calculus of One Variable

##### 2: 1.6 Vectors and Vector-Valued Functions

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►Note that $C$ can be given an orientation by means of $\mathbf{c}$.
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###### Green’s Theorem

… ►###### Stokes’s Theorem

… ►###### Gauss’s (or Divergence) Theorem

… ►###### Green’s Theorem (for Volume)

…##### 3: Bille C. Carlson

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►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.
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►Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities.
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##### 4: 1.9 Calculus of a Complex Variable

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###### DeMoivre’s Theorem

… ►###### Cauchy’s Theorem

… ►###### Liouville’s Theorem

… ►###### Mean Value Property

… ►###### Dominated Convergence Theorem

…##### 5: 2.1 Definitions and Elementary Properties

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►(In other words $=$ here really means
$\subseteq $.)
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►For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $\mathbf{S}$ and $f(z)=O\left({z}^{\nu}\right)$ as $z\to \mathrm{\infty}$ in $\mathbf{S}$, $\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 $\mathbf{S}$ 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 $\mathbf{X}$. … ►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 …##### 6: 19.36 Methods of Computation

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►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).
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►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)).
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►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.
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►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.
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