# zeros

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## 1—10 of 177 matching pages

##### 1: 23.13 Zeros
###### §23.13 Zeros
For information on the zeros of $\wp\left(z\right)$ see Eichler and Zagier (1982).
##### 2: 34.10 Zeros
###### §34.10 Zeros
In a $\mathit{3j}$ symbol, if the three angular momenta $j_{1},j_{2},j_{3}$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3j}$ symbol is zero. …Such zeros are called trivial zeros. …Such zeros are called nontrivial zeros. For further information, including examples of nontrivial zeros and extensions to $\mathit{9j}$ symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).
##### 3: 10.58 Zeros
###### §10.58 Zeros
For $n\geq 0$ the $m$th positive zeros of $\mathsf{j}_{n}\left(x\right)$, $\mathsf{j}_{n}'\left(x\right)$, $\mathsf{y}_{n}\left(x\right)$, and $\mathsf{y}_{n}'\left(x\right)$ are denoted by $a_{n,m}$, $a^{\prime}_{n,m}$, $b_{n,m}$, and $b^{\prime}_{n,m}$, respectively, except that for $n=0$ we count $x=0$ as the first zero of $\mathsf{j}_{0}'\left(x\right)$. …
$a_{n,m}=j_{n+\frac{1}{2},m},$
$b_{n,m}=y_{n+\frac{1}{2},m},$
However, there are no simple relations that connect the zeros of the derivatives. …
##### 4: 9.9 Zeros
###### §9.9 Zeros
They are denoted by $a_{k}$, $a^{\prime}_{k}$, $b_{k}$, $b^{\prime}_{k}$, respectively, arranged in ascending order of absolute value for $k=1,2,\ldots.$
##### 5: 13.22 Zeros
###### §13.22 Zeros
Asymptotic approximations to the zeros when the parameters $\kappa$ and/or $\mu$ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. For example, if $\mu(\geq 0)$ is fixed and $\kappa(>0)$ is large, then the $r$th positive zero $\phi_{r}$ of $M_{\kappa,\mu}\left(z\right)$ is given by …where $j_{2\mu,r}$ is the $r$th positive zero of the Bessel function $J_{2\mu}\left(x\right)$10.21(i)). …
##### 6: 6.13 Zeros
###### §6.13 Zeros
The function $\mathrm{Ei}\left(x\right)$ has one real zero $x_{0}$, given by
6.13.1 $x_{0}=0.37250\;74107\;81366\;63446\;19918\;66580\dots.$
$\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ each have an infinite number of positive real zeros, which are denoted by $c_{k}$, $s_{k}$, respectively, arranged in ascending order of absolute value for $k=0,1,2,\dots$. Values of $c_{1}$ and $c_{2}$ to 30D are given by MacLeod (1996b). …
##### 7: 16.9 Zeros
###### §16.9 Zeros
Then ${{}_{p}F_{p}}\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many zeros if and only if the $a_{j}$ can be re-indexed for $j=1,\dots,p$ in such a way that $a_{j}-b_{j}$ is a nonnegative integer. … Then ${{}_{p}F_{p}}\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many real zeros. … For further information on zeros see Hille (1929).
##### 8: 10.42 Zeros
###### §10.42 Zeros
For example, if $\nu$ is real, then the zeros of $I_{\nu}\left(z\right)$ are all complex unless $-2\ell<\nu<-(2\ell-1)$ for some positive integer $\ell$, in which event $I_{\nu}\left(z\right)$ has two real zeros. … The zeros in the sector $-\tfrac{1}{2}\pi\leq\operatorname{ph}z\leq\tfrac{3}{2}\pi$ are their conjugates. … For $z$-zeros of $K_{\nu}\left(z\right)$, with complex $\nu$, see Ferreira and Sesma (2008). …
##### 9: 8.13 Zeros
###### §8.13 Zeros
• (a)

one negative zero $x_{-}(a)$ and no positive zeros when $1-2n;

• (c)

zeros at $a=-n$ when $x=0$.

• As $x$ increases the positive zeros coalesce to form a double zero at ($a_{n}^{*},x_{n}^{*}$). The values of the first six double zeros are given to 5D in Table 8.13.1. …
##### 10: 14.27 Zeros
###### §14.27 Zeros
$P^{\mu}_{\nu}\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P^{\mu}_{\nu}\left(x\pm i0\right)$ has no zeros in the interval $(-\infty,-1)$. For complex zeros of $P^{\mu}_{\nu}\left(z\right)$ see Hobson (1931, §§233, 234, and 238).