maximum modulus

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1: 1.10 Functions of a Complex Variable

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§1.10(v) Maximum-Modulus Principle

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Analytic Functions

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Harmonic Functions

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Schwarz’s Lemma

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2: 19.27 Asymptotic Approximations and Expansions

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19.27.13 $$R_J(x,y,z,p)=\frac{3}{2\sqrt{z}p}\left(\ln \left(\frac{8z}{a+g}\right)-2R_C(1,\frac{p}{z})+O\left(\left(\frac{a}{z}+\frac{a}{p}\right)\ln \frac{p}{a}\right)\right),$$ $\max (x,y)/\min (z,p)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.12, §19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

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19.27.14 $$R_J(x,y,z,p)=\frac{3}{\sqrt{yz}}{R}_C(x,p)-\frac{6}{yz}{R}_G(0,y,z)+O\left(\frac{\sqrt{x+2p}}{yz}\right),$$ $\max (x,p)/\min (y,z)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

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19.27.16 $$R_J(x,y,z,p)=(3/\sqrt{x})R_C({(h+p)}^2,2(b+h)p)+O\left(\frac{1}{x^{3/2}}\ln \frac{x}{b+h}\right),$$ $\max (y,z,p)/x\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\ln z$: principal branch of logarithm function, $b$ and $h$

Permalink:

http://dlmf.nist.gov/19.27.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

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3: 16.21 Differential Equation

… ►where again $\mathit{\vartheta }=zd/dz$. This equation is of order $\max (p,q)$. …

4: 14.21 Definitions and Basic Properties

… ►The generating function expansions (14.7.19) (with $𝖯$ replaced by $P$) and (14.7.22) apply when ; (14.7.21) (with $𝖯$ replaced by $P$) applies when $|h|>\max \left|z\pm {\left(z^2-1\right)}^{1/2}\right|$.

5: 13.10 Integrals

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13.10.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,c,kt)dt=\mathrm{\Gamma }\left(b\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;c;k/z),$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>\max (\mathrm{\Re }k,0)$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10 and Ch.13

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6: 19.9 Inequalities

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19.9.12 $$\max (\sin \varphi ,\varphi \mathrm{\Delta })\le E(\varphi ,k)\le \varphi ,$$

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Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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7: Bibliography M

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Algorithm 779: Fermi-Dirac functions of order $-1/2$, $1/2$, $3/2$, $5/2$ . ACM Trans. Math. Software 24 (1), pp. 1–12.

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Notes:

Double-precision Fortran, maximum accuracy 16D.

External Links:

ISSN 0098-3500, Document, GAMS (module 779; package TOMS), ZentralBlatt

Referenced by:

8th item.

Encodings:

BibTeX

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8: 3.2 Linear Algebra

… ►and back substitution is $x_n=y_n/d_n$, followed by … ►The $p$-norm of a vector $𝐱={[x_1,\mathrm{\dots },x_n]}^{\mathrm{T}}$ is given by … ►

$${\Vert 𝐱\Vert }_{\mathrm{\infty }}=\underset{1\le j\le n}{\max }\left|x_j\right|.$$

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3.2.14 $${\Vert 𝐀\Vert }_p=\underset{𝐱\ne \mathrm{𝟎}}{\max }\frac{{\Vert 𝐀𝐱\Vert }_p}{{\Vert 𝐱\Vert }_p}.$$

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Referenced by:

§3.2(iii)

Permalink:

http://dlmf.nist.gov/3.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §3.2(iii), §3.2 and Ch.3

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$${\Vert 𝐀\Vert }_1=\underset{1\le k\le n}{\max }\sum _{j=1}^n\left|a_{jk}\right|,$$

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9: 1.5 Calculus of Two or More Variables

… ►The function $f(x,y)$ is continuously differentiable if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous, and twice-continuously differentiable if also ${\partial }^{2}f/{\partial x}^2$, ${\partial }^{2}f/{\partial y}^2$, ${\partial }^{2}f/\partial x\partial y$, 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 $+\mathrm{\infty }$, and $f(x,y)$, $\partial f/\partial x$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times [c,d)$. … ►as $\max ((x_{j+1}-x_j)+(y_{k+1}-y_k))\to 0$. …

10: 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 $(1.1789\mathrm{\dots })\times \frac{1}{4}\pi$; compare Figure 6.3.2. Hence if $x=\pi /(2n)$ and $n\to \mathrm{\infty }$, 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. …