# punctured neighborhood

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## 11—20 of 28 matching pages

##### 11: 31.7 Relations to Other Functions

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►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta =K$, $K+\mathrm{i}{K}^{\prime}$, and $\mathrm{i}{K}^{\prime}$, where $K$ and ${K}^{\prime}$ are related to $k$ as in §19.2(ii).

##### 12: 15.10 Hypergeometric Differential Equation

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►They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity.
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►(a) If $c$ equals $n=1,2,3,\mathrm{\dots}$, and $a=1,2,\mathrm{\dots},n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by (15.10.2) with the interpretation (15.2.5) for ${f}_{2}(z)$.
►(b) If $c$ equals $n=1,2,3,\mathrm{\dots}$, and $a\ne 1,2,\mathrm{\dots},n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by $F(a,b;n;z)$ and
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►(c) If the parameter $c$ in the differential equation equals $2-n=0,-1,-2,\mathrm{\dots}$, then fundamental solutions in the neighborhood of $z=0$ are given by ${z}^{n-1}$ times those in (a) and (b), with $a$ and $b$ replaced throughout by $a+n-1$ and $b+n-1$, respectively.
►(d) If $a+b+1-c$ equals $n=1,2,3,\mathrm{\dots}$, or $2-n=0,-1,-2,\mathrm{\dots}$, then fundamental solutions in the neighborhood of $z=1$ are given by those in (a), (b), and (c) with $z$ replaced by $1-z$.
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##### 13: Bibliography

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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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##### 14: 1.13 Differential Equations

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►For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7.
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##### 15: 3.7 Ordinary Differential Equations

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►For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7.
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##### 16: 33.14 Definitions and Basic Properties

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►This includes $\u03f5=0$, hence $f(\u03f5,\mathrm{\ell};r)$ can be expanded in a convergent power series in $\u03f5$ in a neighborhood of $\u03f5=0$ (§33.20(ii)).
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##### 17: 1.5 Calculus of Two or More Variables

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###### Implicit Function Theorem

►If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\partial F/\partial y\ne 0$ at $(a,b)$, then in a*neighborhood*of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and ${g}^{\prime}(x)=-{F}_{x}/{F}_{y}$. …##### 18: 1.14 Integral Transforms

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►Suppose that $f(t)$ is absolutely integrable on $(-\mathrm{\infty},\mathrm{\infty})$ and of bounded variation in a neighborhood of $t=u$ (§1.4(v)).
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►If $f(t)$ is absolutely integrable on $[0,\mathrm{\infty})$ and of bounded variation (§1.4(v)) in a neighborhood of $t=u$, then
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►Suppose the integral (1.14.32) is absolutely convergent on the line $\mathrm{\Re}s=\sigma $ and $f(x)$ is of bounded variation in a neighborhood of $x=u$.
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##### 19: 2.4 Contour Integrals

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(a)
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In a neighborhood of $a$

2.4.11

$$p(t)=p(a)+\sum _{s=0}^{\mathrm{\infty}}{p}_{s}{(t-a)}^{s+\mu},$$

$$q(t)=\sum _{s=0}^{\mathrm{\infty}}{q}_{s}{(t-a)}^{s+\lambda -1},$$

with $\mathrm{\Re}\lambda >0$, $\mu >0$, ${p}_{0}\ne 0$, and the branches of ${(t-a)}^{\lambda}$ and ${(t-a)}^{\mu}$ continuous and constructed with $\mathrm{ph}\left(t-a\right)\to \omega $ as $t\to a$ along $\mathcal{P}$.

##### 20: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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►Let $f:\mathbf{\Omega}\to \u2102$ (a) be

*orthogonally invariant*, so that $f(\mathbf{T})$ is a symmetric function of ${t}_{1},\mathrm{\dots},{t}_{m}$, the eigenvalues of the matrix argument $\mathbf{T}\in \mathbf{\Omega}$; (b) be analytic in ${t}_{1},\mathrm{\dots},{t}_{m}$ in a neighborhood of $\mathbf{T}=\mathbf{0}$; (c) satisfy $f(\mathbf{0})=1$. …