analytically continued

SearchAdvancedHelp

(0.002 seconds)

1—10 of 42 matching pages

1: 33.13 Complex Variable and Parameters

… ►These functions may also be continued analytically to complex values of $\rho$, $\eta$, and $\mathrm{\ell }$. …

2: 28.7 Analytic Continuation of Eigenvalues

… ►As functions of $q$, $a_n\left(q\right)$ and $b_n\left(q\right)$ can be continued analytically in the complex $q$-plane. …

3: 1.10 Functions of a Complex Variable

… ►The function $f_1(z)$ on $D_1$ is said to be analytically continued along the path $z(t)$, $a\le t\le b$, if there is a chain $(f_1,D_1)$, $(f_2,D_2),\mathrm{\dots },(f_n,D_n)$. … ►Then $f(z)$ can be continued analytically across $\mathrm{𝐴𝐵}$ by reflection, that is, …

4: 2.5 Mellin Transform Methods

… ►Similarly, if $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for $I(x)$ for large values of $x$. … ►Furthermore, $ℳf_1\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane, whose singularities are simple poles at $-{\alpha }_s$, $s=0,1,2,\mathrm{\dots }$, with principal part … ►Similarly, if $\kappa =0$ in (2.5.18), then $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane with simple poles at ${\beta }_s$, $s=0,1,2,\mathrm{\dots }$, with principal part …Alternatively, if $\kappa \ne 0$ in (2.5.18), then $ℳh_2\left(z\right)$ can be continued analytically to an entire function. ►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 …

5: 19.7 Connection Formulas

… ►Provided the functions in these identities are correctly analytically continued in the complex $\beta$-plane, then the identities will also hold in the complex $\beta$-plane. …

6: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $ϵ$ and nonnegative integer values of $\mathrm{\ell }$, but they may be continued analytically to complex arguments and order $\mathrm{\ell }$ as indicated in §33.13. …

7: Bibliography J

… ►

Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

ⓘ

External Links:

MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§8.25(iv).

Encodings:

BibTeX

… ►

Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

ⓘ

External Links:

ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.

Encodings:

BibTeX

…

8: 10.20 Uniform Asymptotic Expansions for Large Order

… ►The function $\zeta =\zeta (z)$ given by (10.20.2) and (10.20.3) can be continued analytically to the $z$-plane cut along the negative real axis. …

9: 2.7 Differential Equations

… ►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. …

10: 18.2 General Orthogonal Polynomials

… ►It is assumed throughout this chapter that for each polynomial $p_n(x)$ that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to the closure of $(a,b)$ unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) …