# analytically continued

(0.001 seconds)

## 1—10 of 38 matching pages

##### 1: 33.13 Complex Variable and Parameters
These functions may also be continued analytically to complex values of $\rho$, $\eta$, and $\ell$. …
##### 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 $z(t)$, $a\leq t\leq b$, if there is a chain $(f_{1},D_{1})$, $(f_{2},D_{2}),\dots,(f_{n},D_{n})$. … Then $f(z)$ can be continued analytically across $\mathit{AB}$ by reflection, that is, …
##### 4: 2.5 Mellin Transform Methods
Similarly, if $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)$ and $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)$ 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, $\mathscr{M}\mskip-3.0mu f_{1}\mskip 3.0mu \left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane, whose singularities are simple poles at $-\alpha_{s}$, $s=0,1,2,\dots$, with principal part … Similarly, if $\kappa=0$ in (2.5.18), then $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane with simple poles at $\beta_{s}$, $s=0,1,2,\dots$, with principal part …Alternatively, if $\kappa\neq 0$ in (2.5.18), then $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ can be continued analytically to an entire function. Since $\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(z\right)$ is analytic for $\Re z>-c$ by Table 2.5.1, the analytically-continued $\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)$ allows us to extend the Mellin transform of $h$ via …
##### 5: 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.) …
##### 6: 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. …
##### 7: 33.22 Particle Scattering and Atomic and Molecular Spectra
The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $\epsilon$ and nonnegative integer values of $\ell$, but they may be continued analytically to complex arguments and order $\ell$ as indicated in §33.13. …
##### 8: Bibliography J
• L. Jacobsen, W. B. Jones, and H. Waadeland (1986) 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.
• W. B. Jones and W. J. Thron (1980) Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.
• ##### 9: 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. …
##### 10: 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. …