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1: 23.13 Zeros

§23.13 Zeros

►For information on the zeros of $\mathrm{\wp }\left(z\right)$ see Eichler and Zagier (1982).

2: 34.10 Zeros

§34.10 Zeros

►In a $\mathit{3}j$ 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 $\mathit{3}j$ 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 $\mathit{9}j$ symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).

3: 10.58 Zeros

§10.58 Zeros

►For $n\ge 0$ the $m$th positive zeros of ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, and ${𝗒}_n^{\prime }\left(x\right)$ are denoted by $a_{n,m}$, $a_{n,m}^{\prime }$, $b_{n,m}$, and $b_{n,m}^{\prime }$, respectively, except that for $n=0$ we count $x=0$ as the first zero of ${𝗃}_0^{\prime }\left(x\right)$. … ►

$a_{n,m}=j_{n+\frac{1}{2},m},$

►

$b_{n,m}=y_{n+\frac{1}{2},m},$

… ►However, there are no simple relations that connect the zeros of the derivatives. …

4: 9.9 Zeros

§9.9 Zeros

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►

§9.9(ii) Relation to Modulus and Phase

… ►

§9.9(iv) Asymptotic Expansions

… ►

§9.9(v) Tables

…

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 $\mu (\ge 0)$ is fixed and $\kappa (>0)$ is large, then the $r$th positive zero ${\varphi }_r$ 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 $$x_0=\mathrm{0.37250\hspace{0.33em}74107\hspace{0.33em}81366\hspace{0.33em}63446\hspace{0.33em}19918\hspace{0.33em}66580}\mathrm{\dots }.$$

ⓘ

Symbols:

$x$: real variable

Notes:

For more digits see OEIS Sequence A091723; see also Sloane (2003).

Referenced by:

§6.13

Permalink:

http://dlmf.nist.gov/6.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.13 and Ch.6

► $\mathrm{Ci}\left(x\right)$ 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 $k=0,1,2,\mathrm{\dots }$. Values of $c_1$ and $c_2$ to 30D are given by MacLeod (1996b). …

7: 16.9 Zeros

§16.9 Zeros

… ►Then ${}_p{}^{}F_p^{}(𝐚;𝐛;z)$ has at most finitely many zeros if and only if the $a_j$ can be re-indexed for $j=1,\mathrm{\dots },p$ in such a way that $a_j-b_j$ is a nonnegative integer. … ►Then ${}_p{}^{}F_p^{}(𝐚;𝐛;z)$ 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 for some positive integer $\mathrm{\ell }$, in which event $I_{\nu }\left(z\right)$ has two real zeros. … ►The zeros in the sector $-\frac{1}{2}\pi \le \mathrm{ph}z\le \frac{3}{2}\pi$ 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

… ►

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

►

one negative zero $x_{-}(a)$ and one positive zero $x_{+}(a)$ when .

… ►As $x$ increases the positive zeros coalesce to form a double zero at ($a_n^{\ast },x_n^{\ast }$). The values of the first six double zeros are given to 5D in Table 8.13.1. …

10: 14.27 Zeros

§14.27 Zeros

► $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ (either side of the cut) has exactly one zero in the interval $(-\mathrm{\infty },-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ has no zeros in the interval $(-\mathrm{\infty },-1)$. ►For complex zeros of $P_{\nu }^{\mu }\left(z\right)$ see Hobson (1931, §§233, 234, and 238).