# maximum modulus

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##### 2: 19.27 Asymptotic Approximations and Expansions
19.27.13 $R_{J}\left(x,y,z,p\right)=\frac{3}{2\sqrt{z}p}\left(\ln\left(\frac{8z}{a+g}% \right)-2R_{C}\left(1,\frac{p}{z}\right)+O\left(\left(\frac{a}{z}+\frac{a}{p}% \right)\ln\frac{p}{a}\right)\right),$ $\max(x,y)/\min(z,p)\to 0$.
19.27.14 $R_{J}\left(x,y,z,p\right)=\frac{3}{\sqrt{yz}}R_{C}\left(x,p\right)-\frac{6}{yz% }R_{G}\left(0,y,z\right)+O\left(\frac{\sqrt{x+2p}}{yz}\right),$ $\max(x,p)/\min(y,z)\to 0$.
19.27.16 $R_{J}\left(x,y,z,p\right)=(3/\sqrt{x})R_{C}\left((h+p)^{2},2(b+h)p\right)+O% \left(\frac{1}{x^{3/2}}\ln\frac{x}{b+h}\right),$ $\max(y,z,p)/x\to 0$.
##### 3: 16.21 Differential Equation
where again $\vartheta=z\ifrac{\mathrm{d}}{\mathrm{d}z}$. This equation is of order $\max(p,q)$. …
##### 4: 14.21 Definitions and Basic Properties
The generating function expansions (14.7.19) (with $\mathsf{P}$ replaced by $P$) and (14.7.22) apply when $|h|<\min\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$; (14.7.21) (with $\mathsf{P}$ replaced by $P$) applies when $|h|>\max\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$.
##### 5: 13.10 Integrals
13.10.3 $\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,c,kt\right)\mathrm{d}t=% \Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;\ifrac{k}{z}% \right),$ $\Re b>0$, $\Re z>\max\left(\Re k,0\right)$,
##### 6: 19.9 Inequalities
19.9.12 $\max(\sin\phi,\phi\Delta)\leq E\left(\phi,k\right)\leq\phi,$
##### 7: Bibliography M
• A. J. MacLeod (1998) Algorithm 779: Fermi-Dirac functions of order $-1/2$, $1/2$, $3/2$, $5/2$ . ACM Trans. Math. Software 24 (1), pp. 1–12.
• ##### 8: 3.2 Linear Algebra
and back substitution is $x_{n}=y_{n}/d_{n}$, followed by … The $p$-norm of a vector $\mathbf{x}=[x_{1},\dots,x_{n}]^{\rm T}$ is given by …
$\|\mathbf{x}\|_{\infty}=\max_{1\leq j\leq n}\left|x_{j}\right|.$
3.2.14 $\|\mathbf{A}\|_{p}=\max_{\mathbf{x}\neq\boldsymbol{{0}}}\frac{\|\mathbf{A}% \mathbf{x}\|_{p}}{\|\mathbf{x}\|_{p}}\,.$
$\|\mathbf{A}\|_{1}=\max_{1\leq k\leq n}\sum_{j=1}^{n}\left|a_{jk}\right|,$
##### 9: 6.16 Mathematical Applications
The first maximum of $\frac{1}{2}\mathrm{Si}\left(x\right)$ for positive $x$ occurs at $x=\pi$ and equals $(1.1789\dots)\times\frac{1}{4}\pi$; compare Figure 6.3.2. Hence if $x=\pi/(2n)$ and $n\to\infty$, then the limiting value of $S_{n}(x)$ overshoots $\frac{1}{4}\pi$ by approximately 18%. Similarly if $x=\pi/n$, then the limiting value of $S_{n}(x)$ undershoots $\frac{1}{4}\pi$ by approximately 10%, and so on. …
##### 10: 1.5 Calculus of Two or More Variables
The function $f(x,y)$ is continuously differentiable if $f$, $\ifrac{\partial f}{\partial x}$, and $\ifrac{\partial f}{\partial y}$ are continuous, and twice-continuously differentiable if also $\ifrac{{\partial}^{2}f}{{\partial x}^{2}}$, $\ifrac{{\partial}^{2}f}{{\partial y}^{2}}$, ${\partial}^{2}f/\partial x\partial y$, and ${\partial}^{2}f/\partial y\partial x$ are continuous. … where $f$ and its partial derivatives on the right-hand side are evaluated at $(a,b)$, and $R_{n}/(\lambda^{2}+\mu^{2})^{n/2}\to 0$ as $(\lambda,\mu)\to(0,0)$. $f(x,y)$ has a local minimum (maximum) at $(a,b)$ if … Suppose that $a,b,c$ are finite, $d$ is finite or $+\infty$, and $f(x,y)$, $\ifrac{\partial f}{\partial x}$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times[c,d)$. … as $\max((x_{j+1}-x_{j})+(y_{k+1}-y_{k}))\to 0$. …