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11: 3.7 Ordinary Differential Equations

… ►where $f$, $g$, and $h$ are analytic functions in a domain $D\subset ℂ$. … ►Assume that we wish to integrate (3.7.1) along a finite path $𝒫$ from $z=a$ to $z=b$ in a domain $D$. The path is partitioned at $P+1$ points labeled successively $z_0,z_1,\mathrm{\dots },z_P$, with $z_0=a$, $z_P=b$. …

12: 2.5 Mellin Transform Methods

… ►The domain of analyticity of $ℳf\left(z\right)$ is usually an infinite strip parallel to the imaginary axis. … ►

Table 2.5.1: Domains of convergence for Mellin transforms.

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Transform	Domain of Convergence
…

►

… ►Next from Table 2.5.1 we observe that the integrals for the transform pair $ℳf_j\left(1-z\right)$ and $ℳh_k\left(z\right)$ are absolutely convergent in the domain $D_{jk}$ specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20). ►

Table 2.5.2: Domains of analyticity for Mellin transforms.

► ►►

Transform Pair	Domain $D_{jk}$
…

►

… ►From Table 2.5.2, we see that each $G_{jk}(z)$ is analytic in the domain $D_{jk}$. …

13: 10.20 Uniform Asymptotic Expansions for Large Order

… ►The eye-shaped closed domain in the uncut $z$-plane that is bounded by $B{P}_1{E}_1$ and $B{P}_2{E}_2$ is denoted by $𝐊$; see Figure 10.20.3. … ►

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Figure 10.20.3: $z$-plane. Domain $𝐊$ (unshaded). … Magnify

…

14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►More generally, a linear operator $T$ on $V$ needs not be defined on all of $V$, but only on a linear subspace $𝒟(T)$ of $V$ which is called the domain of $T$. … ►If $T$ is an unbounded linear operator on a Hilbert space $V$ with dense domain $𝒟(T)$ then the adjoint $T^{\ast }$ of $T$ is the linear operator with domain … ►A linear operator $T$ with dense domain is called symmetric if … … ►Such an operator $T$ is called injective if, for any $u,v$ in its domain, $Tu=Tv$ implies that $u=v$. …

15: Preface

… ►The associate editors are eminent domain experts who were recruited to advise the project on strategy, execution, subject content, format, and presentation, and to help identify and recruit suitable candidate authors and validators. …

16: 13.9 Zeros

… ►Let $P_{\alpha }$ denote the closure of the domain that is bounded by the parabola $y^2=4\alpha (x+\alpha )$ and contains the origin. …

17: Bibliography J

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On the computation of incomplete gamma functions in the complex domain. J. Comput. Appl. Math. 12/13, pp. 401–417.

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External Links:

ISSN 0377-0427, Document, MathReview (M. Marsden), ZentralBlatt

Referenced by:

§8.25(iv), §8.9.

Encodings:

BibTeX

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18: Bibliography N

… ►

The Boutroux ansatz for the second Painlevé equation in the complex domain. Izv. Akad. Nauk SSSR Ser. Mat. 54 (6), pp. 1229–1251 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv. 37(1991), no. 3, pp. 587–609.

External Links:

ISSN 0373-2436, Document, MathReview (Alexander I. Bobenko), ZentralBlatt

Referenced by:

§32.12(ii).

Encodings:

BibTeX

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19: Bibliography K

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Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.

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External Links:

ISSN 0377-9017, Document, MathReview (B. L. J. Braaksma), ZentralBlatt

Referenced by:

§32.11(i), §32.12(i).

Encodings:

BibTeX

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Calculation of modified Bessel functions in a complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 24 (5), pp. 650–664.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(3), pp. 15–24

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.42, §10.74(iv).

Encodings:

BibTeX

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Askey-Wilson Polynomials for Root Systems of Type $BC$ . In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.

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External Links:

FullText, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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20: 3.3 Interpolation

… ►If $f$ is analytic in a simply-connected domain $D$ (§1.13(i)), then for $z\in D$, … ►If $f$ is analytic in a simply-connected domain $D$, then for $z\in D$, …