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11: 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).

12: 10.21 Zeros

§10.21 Zeros

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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

… ► … ►The zeros of the functions …

13: 10.70 Zeros

§10.70 Zeros

►Asymptotic approximations for large zeros are as follows. … ►

$$\text{<span class="ltx\_text hilite">zeros</span> of }{\mathrm{ber}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu -\frac{3}{8})\pi$,

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$$\text{<span class="ltx\_text hilite">zeros</span> of }{\mathrm{bei}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu +\frac{1}{8})\pi$,

… ►In the case $\nu =0$, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the $m$th zero of the function on the left-hand side. …

14: 28.9 Zeros

§28.9 Zeros

►For real $q$ each of the functions ${\mathrm{ce}}_{2n}(z,q)$, ${\mathrm{se}}_{2n+1}(z,q)$, ${\mathrm{ce}}_{2n+1}(z,q)$, and ${\mathrm{se}}_{2n+2}(z,q)$ has exactly $n$ zeros in . …For $q\to \mathrm{\infty }$ the zeros of ${\mathrm{ce}}_{2n}(z,q)$ and ${\mathrm{se}}_{2n+1}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n}\left(q^{1/4}(\pi -2z)\right)$, and the zeros of ${\mathrm{ce}}_{2n+1}(z,q)$ and ${\mathrm{se}}_{2n+2}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n+1}\left(q^{1/4}(\pi -2z)\right)$. …Furthermore, for $q>0$ ${\mathrm{ce}}_m(z,q)$ and ${\mathrm{se}}_m(z,q)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of $J_m\left(2\sqrt{q}\cos z\right)$ (§10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\mathrm{\Im }z\right|\to \mathrm{\infty }$. There are no zeros within the strip other than those on the real and imaginary axes. …

15: 7.13 Zeros

§7.13 Zeros

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Table 7.13.1: Zeros $x_n+\mathrm{i}{y}_n$ of $\mathrm{erf}z.$

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$n$	$x_n$	$y_n$
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Table 7.13.2: Zeros $x_n+\mathrm{i}{y}_n$ of $\mathrm{erfc}z.$

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$n$	$x_n$	$y_n$
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§7.13(iii) Zeros of the Fresnel Integrals

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16: 36.7 Zeros

§36.7 Zeros

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Table 36.7.1: Zeros of cusp diffraction catastrophe to 5D.

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Zeros $\left\{\genfrac{}{}{0.0pt}{}{x}{y}\right\}$ inside, and zeros $\left[\genfrac{}{}{0.0pt}{}{x}{y}\right]$ outside, the cusp $x^2=\frac{8}{27}{|y|}^3$.
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… ►The zeros are approximated by solutions of the equation …

17: 25.18 Methods of Computation

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§25.18(i) Function Values and Derivatives

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§25.18(ii) Zeros

►Most numerical calculations of the Riemann zeta function are concerned with locating zeros of $\zeta \left(\frac{1}{2}+\mathrm{i}t\right)$ in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of $\zeta \left(s\right)$ lie on the critical line $\mathrm{\Re }s=\frac{1}{2}$. Calculations to date (2008) have found no nontrivial zeros off the critical line. …

18: 10.75 Tables

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§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives

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Real Zeros

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Complex Zeros

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§10.75(vi) Zeros of Modified Bessel Functions and their Derivatives

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§10.75(xii) Zeros of Kelvin Functions and their Derivatives

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19: 25.17 Physical Applications

§25.17 Physical Applications

►Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. …

20: 25.10 Zeros

§25.10 Zeros

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§25.10(i) Distribution

… ►These are called the trivial zeros. … ►Calculations relating to the zeros on the critical line make use of the real-valued function …