# punctured neighborhood

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## 11—20 of 28 matching pages

##### 11: 31.7 Relations to Other Functions
Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta=K$, $K+i{K^{\prime}}$, and $i{K^{\prime}}$, where $K$ and ${K^{\prime}}$ are related to $k$ as in §19.2(ii).
##### 12: 15.10 Hypergeometric Differential Equation
They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity. … (a) If $c$ equals $n=1,2,3,\dots$, and $a=1,2,\dots,n-1$, 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 $n=1,2,3,\dots$, and $a\neq 1,2,\dots,n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by $F\left(a,b;n;z\right)$ and … (c) If the parameter $c$ in the differential equation equals $2-n=0,-1,-2,\dots$, 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 $n=1,2,3,\dots$, or $2-n=0,-1,-2,\dots$, then fundamental solutions in the neighborhood of $z=1$ are given by those in (a), (b), and (c) with $z$ replaced by $1-z$. …
##### 13: Bibliography
• V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
• ##### 14: 1.13 Differential Equations
For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
##### 15: 3.7 Ordinary Differential Equations
For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
##### 16: 33.14 Definitions and Basic Properties
This includes $\epsilon=0$, hence $f\left(\epsilon,\ell;r\right)$ can be expanded in a convergent power series in $\epsilon$ in a neighborhood of $\epsilon=0$33.20(ii)). …
##### 17: 1.5 Calculus of Two or More Variables
###### Implicit Function Theorem
If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\ifrac{\partial F}{\partial y}\not=0$ 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
Suppose that $f(t)$ is absolutely integrable on $(-\infty,\infty)$ and of bounded variation in a neighborhood of $t=u$1.4(v)). … If $f(t)$ is absolutely integrable on $[0,\infty)$ and of bounded variation (§1.4(v)) in a neighborhood of $t=u$, then … Suppose the integral (1.14.32) is absolutely convergent on the line $\Re s=\sigma$ and $f(x)$ is of bounded variation in a neighborhood of $x=u$. …
##### 19: 2.4 Contour Integrals
• (a)

In a neighborhood of $a$

with $\Re\lambda>0$, $\mu>0$, $p_{0}\neq 0$, and the branches of $(t-a)^{\lambda}$ and $(t-a)^{\mu}$ continuous and constructed with $\operatorname{ph}\left(t-a\right)\to\omega$ as $t\to a$ along $\mathscr{P}$.

• ##### 20: 35.7 Gaussian Hypergeometric Function of Matrix Argument
Let $f:{\boldsymbol{\Omega}}\to\mathbb{C}$ (a) be orthogonally invariant, so that $f(\mathbf{T})$ is a symmetric function of $t_{1},\dots,t_{m}$, the eigenvalues of the matrix argument $\mathbf{T}\in{\boldsymbol{\Omega}}$; (b) be analytic in $t_{1},\dots,t_{m}$ in a neighborhood of $\mathbf{T}=\boldsymbol{{0}}$; (c) satisfy $f(\boldsymbol{{0}})=1$. …