analytically continued

SearchAdvancedHelp

(0.002 seconds)

31—40 of 42 matching pages

31: Errata

… ►It was decided that much more information should be given in the section on general OP’s, and as a consequence Chapter 1 (Algebraic and Analytic Methods), also required a significant expansion. … ►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. … ►

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, ${𝖯}_n,{𝖰}_n,P_n,Q_n,{𝑸}_n$ and the Laguerre polynomial, $L_n$, were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

… ►

A new Subsection Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

… ►

A new Subsection Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

…

32: 1.9 Calculus of a Complex Variable

… ►

§1.9(ii) Continuity, Point Sets, and Differentiation

… ►Differentiability automatically implies continuity. … ►

Analyticity

… ► … ► …

33: 30.3 Eigenvalues

… ►The eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are analytic functions of the real variable ${\gamma }^2$ and satisfy … ►

§30.3(iii) Transcendental Equation

►If $p$ is an even nonnegative integer, then the continued-fraction equation …If $p=0$ or $p=1$, the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if $p>1$ its last denominator is ${\beta }_0-\lambda$ or ${\beta }_1-\lambda$. …

34: 7.18 Repeated Integrals of the Complementary Error Function

… ►The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors one has to use the analytic continuation formula (13.2.12). … ►

§7.18(v) Continued Fraction

…

35: 1.4 Calculus of One Variable

… ►

§1.4(ii) Continuity

… ►When $n\ge 1$, $f$ is continuously differentiable on $I$. … … ►Continuity, or piecewise continuity, of $f(x)$ on $[a,b]$ is sufficient for the limit to exist. … ►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …

36: Bibliography S

… ►

Use of analyticity in the calculation of nonrelativistic scattering amplitudes. Phys. Rev. 167, pp. 1411–1423.

ⓘ

External Links:

Document, Link

Referenced by:

(18.40.9), §18.40(ii).

Encodings:

BibTeX

… ►

Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

ⓘ

Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

§33.1, §33.14(iv), 9th item.

Encodings:

BibTeX

… ►

A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Comm. 105 (2-3), pp. 263–272.

ⓘ

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

… ►

Exact analytical solutions of $z{e}^z=a$ . J. Math. Anal. Appl. 43 (3), pp. 626–632.

ⓘ

External Links:

Document, MathReview (Pl. Kannappan), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

… ►

Resonances in $n$-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory. Ann. of Math. (2) 97, pp. 247–274.

ⓘ

External Links:

ISSN 0003-486X, Document, Link, MathReview (K. Kumar)

Referenced by:

§1.18(viii).

Encodings:

BibTeX

…

37: Bibliography R

… ►

Topics in Analytic Number Theory. Springer-Verlag, New York.

ⓘ

Notes:

Edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169

External Links:

MathReview (H. Halberstam), ZentralBlatt

Referenced by:

§1.8(iv), §20.7(viii), §20.7(viii), Ch.24.

Encodings:

BibTeX

… ►

A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

…

38: 2.11 Remainder Terms; Stokes Phenomenon

… ►when $\mathrm{\Re }p>0$ and , and by analytic continuation for other values of $p$ and $z$. … ►Following §2.4(iv), we rotate the integration path through an angle $-\theta$, which is valid by analytic continuation when . … ►That the change in their forms is discontinuous, even though the function being approximated is analytic, is an example of the Stokes phenomenon. … ►

§2.11(v) Exponentially-Improved Expansions (continued)

…

39: Bibliography M

… ►

The Theory of Analytic Functions: A Brief Course. “Mir”, Moscow.

ⓘ

Notes:

Translated from the Russian by Eugene Yankovsky

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.10(iii), §1.10(vi), §1.9(i), §1.9(iv), §1.9(vi).

Encodings:

BibTeX

… ►

Analytic expressions for integrals of products of spherical Bessel functions. J. Phys. A 24 (7), pp. 1435–1453.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.59.

Encodings:

BibTeX

… ►

Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

… ►

A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

ⓘ

External Links:

ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

… ►

A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.

ⓘ

External Links:

ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

…

40: 1.14 Integral Transforms

… ►

Convergence and Analyticity

… ►Then $ℒf\left(s\right)$ is an analytic function of $s$ for $\mathrm{\Re }s>\alpha$. … ►If $x^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty })$ for all $\sigma$ in , then the integral (1.14.32) converges and $ℳf\left(s\right)$ is an analytic function of $s$ in the vertical strip . … ►Suppose $f(t)$ is continuously differentiable on $(-\mathrm{\infty },\mathrm{\infty })$ and vanishes outside a bounded interval. … ►In this case, $𝒮f\left(s\right)$ represents an analytic function in the $s$-plane cut along the negative real axis, and …