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21: 1.9 Calculus of a Complex Variable

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Continuity

… ► … ►If $x(t)$ and $y(t)$ are continuous and $x^{\prime }(t)$ and $y^{\prime }(t)$ are piecewise continuous, then $z(t)$ defines a contour. … ►for a contour $C$ and $f(z(t))$ continuous, $a\le t\le b$. … ►If $f(z)$ is continuous within and on a simple closed contour $C$ and analytic within $C$, then …

22: 2.2 Transcendental Equations

… ►Let $f(x)$ be continuous and strictly increasing when and …

23: 4.12 Generalized Logarithms and Exponentials

… ►For $C^{\mathrm{\infty }}$ generalized logarithms, see Walker (1991). …

24: 14.25 Integral Representations

… ►where the multivalued functions have their principal values when and are continuous in $ℂ\setminus (-\mathrm{\infty },1]$. …

25: Bibliography J

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Continuous Univariate Distributions. 2nd edition, Vol. I, John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-58495-9, MathReview (D. N. Shanbhag), ZentralBlatt

Referenced by:

§7.20(iii), §8.23.

Encodings:

BibTeX

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Continuous Univariate Distributions. 2nd edition, Vol. II, John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-58494-0, MathReview (D. N. Shanbhag), ZentralBlatt

Referenced by:

§8.23.

Encodings:

BibTeX

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26: 3.7 Ordinary Differential Equations

… ►Let $(a,b)$ be a finite or infinite interval and $q(x)$ be a real-valued continuous (or piecewise continuous) function on the closure of $(a,b)$. … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. … ►The order estimate $O\left(h^5\right)$ holds if the solution $w(z)$ has five continuous derivatives. … ►The order estimates $O\left(h^5\right)$ hold if the solution $w(z)$ has five continuous derivatives. …

27: Bibliography F

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Multivariate Calculation. Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96049-X, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

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Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

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Notes:

Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.15.

Encodings:

BibTeX

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28: 1.14 Integral Transforms

… ►If $f(t)$ is continuous and $f^{\prime }(t)$ is piecewise continuous on $[0,\mathrm{\infty })$, then … ►If $f(t)$ is piecewise continuous, then … ►If $f(t)$ is continuous on $[0,\mathrm{\infty })$ and $f^{\prime }(t)$ is piecewise continuous on $(0,\mathrm{\infty })$, then … ►Next, assume $f(t)$, $f^{\prime }(t)$, $\mathrm{\dots }$, $f^{(n-1)}(t)$ are continuous and each satisfies (1.14.18). … ►If $f(t)$ and $g(t)$ are piecewise continuous, then …

29: 1.10 Functions of a Complex Variable

… ►Also, let $f(z)$ be analytic within $C$, continuous within and on $C$, and real on $\mathrm{𝐴𝐵}$. … ►If $f(z)$ is analytic within a simple closed contour $C$, and continuous within and on $C$—except in both instances for a finite number of singularities within $C$—then … ►If $f(z)$ is continuous on $\overline{D}$ and analytic in $D$, then $\left|f(z)\right|$ attains its maximum on $\partial D$. … ►Moreover, if $D$ is bounded and $u(z)$ is continuous on $\overline{D}$ and harmonic in $D$, then $u(z)$ is maximum at some point on $\partial D$. … ►(b) By specifying the value of $F(z)$ at a point $z_0$ (not a branch point), and requiring $F(z)$ to be continuous on any path that begins at $z_0$ and does not pass through any branch points or other singularities of $F(z)$. …

30: 2.1 Definitions and Elementary Properties

… ►For example, suppose $f(x)$ is continuous and $f(x)\sim x^{\nu }$ as $x\to +\mathrm{\infty }$ in $ℝ$, where $\nu$ ($\in ℂ$) is a constant. … ►

2.1.11 $${\int }_x^{\mathrm{\infty }}f(t)dt\sim -\frac{x^{\nu +1}}{\nu +1},$$ ,

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Symbols:

$\sim$: asymptotic equality, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $f(x)$: continuous function and $\nu$: complex constant

Permalink:

http://dlmf.nist.gov/2.1.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(ii), §2.1 and Ch.2

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2.1.12

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Symbols:

$\sim$: asymptotic equality, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $f(x)$: continuous function and $\nu$: complex constant

Permalink:

http://dlmf.nist.gov/2.1.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.1(ii), §2.1 and Ch.2

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