# analytically continued

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## 1—10 of 42 matching pages

##### 1: 33.13 Complex Variable and Parameters

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►These functions may also be continued analytically to complex values of $\rho $, $\eta $, and $\mathrm{\ell}$.
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##### 2: 28.7 Analytic Continuation of Eigenvalues

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►As functions of $q$, ${a}_{n}\left(q\right)$ and ${b}_{n}\left(q\right)$ can be continued analytically in the complex $q$-plane.
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##### 3: 1.10 Functions of a Complex Variable

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►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 $\mathrm{\mathit{A}\mathit{B}}$ by*reflection*, that is, …##### 4: 2.5 Mellin Transform Methods

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►Similarly, if $\mathcal{M}f\left(1-z\right)$ and $\mathcal{M}h\left(z\right)$ 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 $x$.
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►Furthermore, $\mathcal{M}{f}_{1}\left(z\right)$ 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
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►Similarly, if $\kappa =0$ in (2.5.18), then $\mathcal{M}{h}_{2}\left(z\right)$ 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 $\mathcal{M}{h}_{2}\left(z\right)$ can be continued analytically to an entire function.
►Since $\mathcal{M}{h}_{1}\left(z\right)$ is analytic for $\mathrm{\Re}z>-c$ by Table 2.5.1, the analytically-continued
$\mathcal{M}{h}_{2}\left(z\right)$ allows us to extend the Mellin transform of $h$ via
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##### 5: 19.7 Connection Formulas

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►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.
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##### 6: 33.22 Particle Scattering and Atomic and Molecular Spectra

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►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho $, $r$, $\eta $, $\u03f5$ 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.
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##### 7: Bibliography J

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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.
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Continued Fractions: Analytic Theory and Applications.
Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.
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##### 8: 10.20 Uniform Asymptotic Expansions for Large Order

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►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.
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##### 9: 2.7 Differential Equations

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►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.
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►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.
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##### 10: 18.2 General Orthogonal Polynomials

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►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.) …