# circular cases

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## 11—20 of 171 matching pages

##### 11: 6.9 Continued Fraction
6.9.1 $E_{1}\left(z\right)=\cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+% \cfrac{2}{z+\cfrac{3}{1+\cfrac{3}{z+}}}}}}}\cdots,$ $|\operatorname{ph}z|<\pi$.
##### 12: 15.19 Methods of Computation
This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi\mathrm{i}/3},{\mathrm{e}}^{-\pi\mathrm{i}/3}\}$ onto itself. …
##### 13: 10.37 Inequalities; Monotonicity
If $0\leq\nu<\mu$ and $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$, then …Note that previously we did mention that (10.37.1) holds for $|\operatorname{ph}z|<\pi$. This is definitely not the case. …
##### 14: 1.10 Functions of a Complex Variable
If $D=\mathbb{C}\setminus(-\infty,0]$ and $z=re^{i\theta}$, then one branch is $\sqrt{r}e^{i\theta/2}$, the other branch is $-\sqrt{r}e^{i\theta/2}$, with $-\pi<\theta<\pi$ in both cases. Similarly if $D=\mathbb{C}\setminus[0,\infty)$, then one branch is $\sqrt{r}e^{i\theta/2}$, the other branch is $-\sqrt{r}e^{i\theta/2}$, with $0<\theta<2\pi$ in both cases. …
##### 15: 19.6 Special Cases
Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_{C}\left(x,y\right)$. …
##### 16: 16.5 Integral Representations and Integrals
In the case $p=q$ the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when $|\operatorname{ph}\left(-z\right)|<\pi/2$. In the case $p=q+1$ the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector $|\operatorname{ph}\left(1-z\right)|<\pi$; compare §16.2(iii). …
##### 17: 22.5 Special Values
For values of $K,K^{\prime}$ when $k^{2}=\frac{1}{2}$ (lemniscatic case) see §23.5(iii), and for $k^{2}=e^{\mathrm{i}\pi/3}$ (equianharmonic case) see §23.5(v). …
##### 18: 15.9 Relations to Other Functions
15.9.19 $\mathbf{F}\left({a,b\atop a-b+1};z\right)=z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*P^{b% -a}_{-b}\left(\frac{1+z}{1-z}\right),$ $|\operatorname{ph}z|<\pi$ and $|\operatorname{ph}\left(1-z\right)|<\pi$.
For the case $0 see (14.3.1). …
##### 19: 10.41 Asymptotic Expansions for Large Order
We then extend the validity of this property from $z\to\pm i\infty$ to $z\to\infty$ in the sector $-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta$ in the case of ${H^{(1)}_{\nu}}\left(\nu z\right)$, and to $z\to\infty$ in the sector $-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta$ in the case of ${H^{(2)}_{\nu}}\left(\nu z\right)$. …
##### 20: 28.29 Definitions and Basic Properties
The case $c=0$ is equivalent to …The solutions of period $\pi$ or $2\pi$ are exceptional in the following sense. … In the symmetric case $Q(z)=Q(-z)$, $w_{\mbox{\tiny I}}(z,\lambda)$ is an even solution and $w_{\mbox{\tiny II}}(z,\lambda)$ is an odd solution; compare §28.2(ii). …The cases $\nu=0$ and $\nu=1$ split into four subcases as in (28.2.21) and (28.2.22). …