analyticity

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41: Tom M. Apostol

… ►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects. …

42: Richard A. Askey

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1 Algebraic and Analytic Methods

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43: 4.2 Definitions

… ►This is a multivalued function of $z$ with branch point at $z=0$. … ► $\ln z$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },0]$ and real-valued when $z$ ranges over the positive real numbers. … ►

§4.2(iii) The Exponential Function

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4.2.27 $$z^a=\underset{n\text{ times}}{\underbrace{z\cdot z\mathrm{\cdots }z}}=1/z^{-a}.$$

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Defines:

$a=n$: integer (locally)

Symbols:

$n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

… ►This is an analytic function of $z$ on $ℂ\setminus (-\mathrm{\infty },0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$. …

44: 4.28 Definitions and Periodicity

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Periodicity and Zeros

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45: 22.17 Moduli Outside the Interval [0,1]

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§22.17(ii) Complex Moduli

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46: 10.11 Analytic Continuation

§10.11 Analytic Continuation

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47: 36.12 Uniform Approximation of Integrals

… ►Also, $f$ is real analytic, and ${\partial }^{K+2}f/{\partial u}^{K+2}>0$ for all $𝐲$ such that all $K+1$ critical points coincide. If , then we may evaluate the complex conjugate of $I$ for real values of $𝐲$ and $g$, and obtain $I$ by conjugation and analytic continuation. … ►The square roots are real and positive when $𝐲$ is such that all the critical points are real, and are defined by analytic continuation elsewhere. The quantities $a_m(𝐲)$ are real for real $𝐲$ when $g$ is real analytic. … ►The coefficients of $\mathrm{Ai}$ and ${\mathrm{Ai}}^{\prime }$ are real if $y$ is real and $g$ is real analytic. …

48: 2.7 Differential Equations

… ►is one at which the coefficients $f(z)$ and $g(z)$ are analytic. All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►If both $(z-z_0)f(z)$ and ${(z-z_0)}^{2}g(z)$ are analytic at $z_0$, then $z_0$ is a regular singularity (or singularity of the first kind). … ►Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution $w_j(z)$ can be continued analytically into any other sector. … ►In a finite or infinite interval $(a_1,a_2)$ let $f(x)$ be real, positive, and twice-continuously differentiable, and $g(x)$ be continuous. …

49: 2.5 Mellin Transform Methods

… ►With these definitions and the conditions (2.5.17)–(2.5.20) the Mellin transforms converge absolutely and define analytic functions in the half-planes shown in Table 2.5.1. … ►Since $ℳh_1\left(z\right)$ is analytic for $\mathrm{\Re }z>-c$ by Table 2.5.1, the analytically-continued $ℳh_2\left(z\right)$ allows us to extend the Mellin transform of $h$ via … ►From Table 2.5.2, we see that each $G_{jk}(z)$ is analytic in the domain $D_{jk}$. Furthermore, each $G_{jk}(z)$ has an analytic or meromorphic extension to a half-plane containing $D_{jk}$. … ►Since $ℳh\left(z\right)$ is analytic for $\mathrm{\Re }z>-c$, by (2.5.14), …

50: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a_k+s\right)$, $k=1,\mathrm{\dots },p$, from those of $\mathrm{\Gamma }\left(-s\right)$. … ►In the case $p=q+1$ the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector ; compare §16.2(iii). …