analytically continued

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21: 31.11 Expansions in Series of Hypergeometric Functions

… ►For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\mathrm{\infty }$. …In this case the accessory parameter $q$ is a root of the continued-fraction equation …

22: 13.2 Definitions and Basic Properties

… ► ►Although $M(a,b,z)$ does not exist when $b=-n$, $n=0,1,2,\mathrm{\dots }$, many formulas containing $M(a,b,z)$ continue to apply in their limiting form. … ►

§13.2(ii) Analytic Continuation

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23: 14.21 Definitions and Basic Properties

… ►When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). … …

24: 13.14 Definitions and Basic Properties

… ► ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►

§13.14(ii) Analytic Continuation

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25: 4.23 Inverse Trigonometric Functions

… ►

4.23.6 $$\mathrm{Arccot}z=\mathrm{Arctan}\left(1/z\right).$$

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Defines:

$\mathrm{Arccot}z$: general arccotangent function

Symbols:

$\mathrm{Arctan}z$: general arctangent function and $z$: complex variable

A&S Ref:

4.4.8 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

… ►The function ${(1-t^2)}^{1/2}$ assumes its principal value when $t\in (-1,1)$; elsewhere on the integration paths the branch is determined by continuity. … ►These functions are analytic in the cut plane depicted in Figures 4.23.1(iii) and 4.23.1(iv). …

26: 2.10 Sums and Sequences

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On the strip $a\le \mathrm{\Re }z\le n$, $f(z)$ is analytic in its interior, $f^{(2m)}(z)$ is continuous on its closure, and $f(z)=o\left({\mathrm{e}}^{2\pi |\mathrm{\Im }z|}\right)$ as $\mathrm{\Im }z\to \pm \mathrm{\infty }$, uniformly with respect to $\mathrm{\Re }z\in [a,n]$.

… ►Let $f(z)$ be analytic on the annulus , with Laurent expansion … ►

$g(z)$ is analytic on .

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On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_j$, say,

2.10.30 $$f(z)-g(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$ $z\to z_j$,

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant

Permalink:

http://dlmf.nist.gov/2.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iv), §2.10 and Ch.2

where ${\sigma }_j$ is a positive constant.

►Secondly, when $f(z)-g(z)$ is $m$ times continuously differentiable on $|z|=r$ the result (2.10.29) can be strengthened. …

27: 8.19 Generalized Exponential Integral

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§8.19(vii) Continued Fraction

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§8.19(viii) Analytic Continuation

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28: 3.3 Interpolation

… ►If $f$ is analytic in a simply-connected domain $D$ (§1.13(i)), then for $z\in D$, … ►If $f$ and the $z_k$ ($=x_k$) are real, and $f$ is $n$ times continuously differentiable on a closed interval containing the $x_k$, then …If $f$ is analytic in a simply-connected domain $D$, then for $z\in D$, … ►For Hermite interpolation, trigonometric interpolation, spline interpolation, rational interpolation (by using continued fractions), interpolation based on Chebyshev points, and bivariate interpolation, see Bulirsch and Rutishauser (1968), Davis (1975, pp. 27–31), and Mason and Handscomb (2003, Chapter 6). …

29: 31.4 Solutions Analytic at Two Singularities: Heun Functions

§31.4 Solutions Analytic at Two Singularities: Heun Functions

►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . … ►The eigenvalues $q_m$ satisfy the continued-fraction equation … ►with $(s_1,s_2)\in \{0,1,a,\mathrm{\infty }\}$, denotes a set of solutions of (31.2.1), each of which is analytic at $s_1$ and $s_2$. …

30: 2.4 Contour Integrals

… ►If $q(t)$ is analytic in a sector containing $\mathrm{ph}t=0$, then the region of validity may be increased by rotation of the integration paths. …(The branches of $t^{(s+\lambda -\mu )/\mu }$ and $z^{(s+\lambda )/\mu }$ are extended by continuity.) … ►is continuous in $\mathrm{\Re }z\ge c$ and analytic in $\mathrm{\Re }z>c$, and by inversion (§1.14(iii)) … ►Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion … ►Assume that $p(t)$ and $q(t)$ are analytic on an open domain $𝐓$ that contains $𝒫$, with the possible exceptions of $t=a$ and $t=b$. …