# mean value theorems

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##### 2: 1.6 Vectors and Vector-Valued Functions
Note that $C$ can be given an orientation by means of $\mathbf{c}$. …
##### 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. …
##### 5: 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 $\mathbf{S}$ and $f(z)=O\left(z^{\nu}\right)$ as $z\to\infty$ in $\mathbf{S}$, $\nu$ being real, then $f^{\prime}(z)=O\left(z^{\nu-1}\right)$ as $z\to\infty$ 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 $|\operatorname{ph}z|\leq\frac{1}{2}\pi-\delta$ ($<\frac{1}{2}\pi$) each of the functions $0,e^{-z}$, and $e^{-\sqrt{z}}$ (principal value) has the null asymptotic expansion …
##### 6: 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 $F\left(\phi,k\right)$ when $\frac{1}{2}\pi-\phi$ is small and positive. …These special theorems are also useful for checking computer codes. …
##### 7: 10.18 Modulus and Phase Functions
10.18.17 ${M_{\nu}}^{2}\left(x\right)\sim\frac{2}{\pi x}\left(1+\frac{1}{2}\frac{\mu-1}{% (2x)^{2}}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu-1)(\mu-9)}{(2x)^{4}}+\frac{1% \cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{(\mu-1)(\mu-9)(\mu-25)}{(2x)^{6}}+\dotsb% \right),$
10.18.18 $\theta_{\nu}\left(x\right)\sim x-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi+% \frac{\mu-1}{2(4x)}+\frac{(\mu-1)(\mu-25)}{6(4x)^{3}}+\frac{(\mu-1)(\mu^{2}-11% 4\mu+1073)}{5(4x)^{5}}+\frac{(\mu-1)(5\mu^{3}-1535\mu^{2}+54703\mu-3\;75733)}{% 14(4x)^{7}}+\dotsb.$
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 $-\frac{3}{2}\leq\nu\leq\frac{3}{2}$ for (10.18.18).