# as z→0

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## 11—20 of 609 matching pages

##### 11: 22.5 Special Values
For example, at $z=K+iK^{\prime}$, $\operatorname{sn}\left(z,k\right)=1/k$, $\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0$. …
##### 12: 16.8 Differential Equations
is a value $z_{0}$ of $z$ at which all the coefficients $f_{j}(z)$, $j=0,1,\dots,n-1$, are analytic. If $z_{0}$ is not an ordinary point but $(z-z_{0})^{n-j}f_{j}(z)$, $j=0,1,\dots,n-1$, are analytic at $z=z_{0}$, then $z_{0}$ is a regular singularity. … Equation (16.8.4) has a regular singularity at $z=0$, and an irregular singularity at $z=\infty$, whereas (16.8.5) has regular singularities at $z=0$, $1$, and $\infty$. … More generally if $z_{0}$ ($\in\mathbb{C}$) is an arbitrary constant, $|z-z_{0}|>\max{(|z_{0}|,|z_{0}-1|)}$, and $|\operatorname{ph}\left(z_{0}-z\right)|<\pi$, then …(Note that the generalized hypergeometric functions on the right-hand side are polynomials in $z_{0}$.) …
##### 13: 12.8 Recurrence Relations and Derivatives
12.8.1 $zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0,$
12.8.2 $U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2})U\left(a+1,z% \right)=0,$
12.8.3 $U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)=0,$
12.8.6 $V'\left(a,z\right)-\tfrac{1}{2}zV\left(a,z\right)-(a-\tfrac{1}{2})V\left(a-1,z% \right)=0,$
12.8.7 $V'\left(a,z\right)+\tfrac{1}{2}zV\left(a,z\right)-V\left(a+1,z\right)=0,$
##### 14: 3.3 Interpolation
The divided differences of $f$ relative to a sequence of distinct points $z_{0},z_{1},z_{2},\dots$ are defined by … Newton’s formula has the advantage of allowing easy updating: incorporation of a new point $z_{n+1}$ requires only addition of the term with $[z_{0},z_{1},\dots,z_{n+1}]f$ to (3.3.38), plus the computation of this divided difference. Another advantage is its robustness with respect to confluence of the set of points $z_{0},z_{1},\dots,z_{n}$. For example, for $k+1$ coincident points the limiting form is given by $[z_{0},z_{0},\dots,z_{0}]f=f^{(k)}(z_{0})/k!$. … It can be used for solving a nonlinear scalar equation $f(z)=0$ approximately. …
##### 15: 3.7 Ordinary Differential Equations
The path is partitioned at $P+1$ points labeled successively $z_{0},z_{1},\dots,z_{P}$, with $z_{0}=a$, $z_{P}=b$. …
$f_{0}(z)=1,$
$g_{0}(z)=0,$
$h_{0}(z)=0,$
$f_{1}(z)=0,$
##### 16: 10.35 Generating Function and Associated Series
For $z\in\mathbb{C}$ and $t\in\mathbb{C}\setminus\{0\}$, …
10.35.4 $1=I_{0}\left(z\right)-2I_{2}\left(z\right)+2I_{4}\left(z\right)-2I_{6}\left(z% \right)+\dotsb,$
10.35.5 $e^{\pm z}=I_{0}\left(z\right)\pm 2I_{1}\left(z\right)+2I_{2}\left(z\right)\pm 2% I_{3}\left(z\right)+\dotsb,$
$\cosh z=I_{0}\left(z\right)+2I_{2}\left(z\right)+2I_{4}\left(z\right)+2I_{6}% \left(z\right)+\dots,$
##### 17: 32.7 Bäcklund Transformations
with $\zeta=-2^{1/3}z$ and $\varepsilon=\pm 1$, where $W(\zeta;\tfrac{1}{2}\varepsilon)$ satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ with $z=\zeta$, $\alpha=\tfrac{1}{2}\varepsilon$, and $w(z;0)$ satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$. … Let $w_{0}=w(z;\alpha_{0},\beta_{0})$ and $w_{j}^{\pm}=w(z;\alpha_{j}^{\pm},\beta_{j}^{\pm})$, $j=1,2,3,4$, be solutions of $\mbox{P}_{\mbox{\scriptsize IV}}$ with …
$z_{1}=-z_{0},$
$z_{2}=z_{0},$
Let $W_{0}=W(z;\alpha_{0},\beta_{0},\gamma_{0},-\tfrac{1}{2})$ and $W_{1}=W(z;\alpha_{1},\beta_{1},\gamma_{1},-\tfrac{1}{2})$ be solutions of $\mbox{P}_{\mbox{\scriptsize V}}$, where …
##### 18: 19.21 Connection Formulas
Upper signs apply if $0<\operatorname{ph}z<\pi$, and lower signs if $-\pi<\operatorname{ph}z<0$:
19.21.4 $R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)\mp\mathrm{i}R_{F}\left(0,z% ,1\right),$
If $0 and $y=z+1$, then as $p\to 0$ (19.21.6) reduces to Legendre’s relation (19.21.1). … Because $R_{G}$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\leq 0$ if the variables are real, thereby avoiding cancellations when $R_{G}$ is calculated from $R_{F}$ and $R_{D}$ (see §19.36(i)). … Let $x,y,z$ be real and nonnegative, with at most one of them 0. …
##### 19: 1.9 Calculus of a Complex Variable
and when $z\neq 0$, … A function $f(z)$ is continuous at a point $z_{0}$ if $\lim\limits_{z\to z_{0}}f(z)=f(z_{0})$. … ($z_{0}$ may or may not belong to $S$.) … A function $f(z)$ is said to be analytic (holomorphic) at $z=z_{0}$ if it is differentiable in a neighborhood of $z_{0}$. … where $\mathcal{N}(C,z_{0})$ is an integer called the winding number of $C$ with respect to $z_{0}$ . …
##### 20: 10.2 Definitions
This solution of (10.2.1) is an analytic function of $z\in\mathbb{C}$, except for a branch point at $z=0$ when $\nu$ is not an integer. … For fixed $z$ $(\neq 0)$ each branch of $J_{\nu}\left(z\right)$ is entire in $\nu$. … Whether or not $\nu$ is an integer $Y_{\nu}\left(z\right)$ has a branch point at $z=0$. … For fixed $z$ $(\neq 0)$ each branch of $Y_{\nu}\left(z\right)$ is entire in $\nu$. … Each solution has a branch point at $z=0$ for all $\nu\in\mathbb{C}$. …