exceptional values

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21: 34.4 Definition: $\mathit{6}j$ Symbol

… ►where the summation is taken over all admissible values of the $m$’s and $m^{\prime }$’s for each of the four $\mathit{3}j$ symbols; compare (34.2.2) and (34.2.3). ►Except in degenerate cases the combination of the triangle inequalities for the four $\mathit{3}j$ symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths $j_1,j_2,j_3,l_1,l_2,l_3$; see Figure 34.4.1. …

22: 28.12 Definitions and Basic Properties

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§28.12(i) Eigenvalues ${\lambda }_{\nu +2n}\left(q\right)$

►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. … ► … ►As in §28.7 values of $q$ for which (28.2.16) has simple roots $\lambda$ are called normal values with respect to $\nu$. … ►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization …

23: 10.2 Definitions

… ►Except in the case of $J_{\pm n}\left(z\right)$, the principal branches of $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$; compare §4.2(i). … ►Except where indicated otherwise, it is assumed throughout the DLMF that the symbols $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$ denote the principal values of these functions. …

24: 4.23 Inverse Trigonometric Functions

… ►Except where indicated otherwise, it is assumed throughout the DLMF that the inverse trigonometric functions assume their principal values. …

25: 10.25 Definitions

… ►Except where indicated otherwise it is assumed throughout the DLMF that the symbols $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ denote the principal values of these functions. …

26: 10.75 Tables

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Makinouchi (1966) tabulates all values of $j_{\nu ,m}$ and $y_{\nu ,m}$ in the interval $(0,100)$, with at least 29S. These are for $\nu =0(1)5$, 10, 20; $\nu =\frac{3}{2}$, $\frac{5}{2}$; $\nu =m/n$ with $m=1(1)n-1$ and $n=3(1)8$, except for $\nu =\frac{1}{2}$.

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27: 19.16 Definitions

… ►In (19.16.1)–(19.16.2_5), $x,y,z\in ℂ\setminus (-\mathrm{\infty },0]$ except that one or more of $x,y,z$ may be 0 when the corresponding integral converges. In (19.16.2) the Cauchy principal value is taken when $p$ is real and negative. …

28: 19.3 Graphics

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Figure 19.3.4: $E(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. If ${\sin }^2\varphi =1$ ($\ge k^2$), then the function reduces to $E\left(k\right)$, with value 1 at $k^2=1$. … Magnify 3D Help

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Figure 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ as a function of $k^2$ and ${\alpha }^2$ for , $-2\le {\alpha }^2\le 2$. Cauchy principal values are shown when ${\alpha }^2>1$. … Magnify 3D Help

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . Cauchy principal values are shown when ${\sin }^2\varphi >\frac{1}{2}$. The function tends to $+\mathrm{\infty }$ as ${\sin }^2\varphi \to \frac{1}{2}$, except in the last case below. … Magnify 3D Help

… ►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. …

29: 15.6 Integral Representations

… ►In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. … ►In (15.6.1) all functions in the integrand assume their principal values. ►In (15.6.2) the point $1/z$ lies outside the integration contour, $t^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty })$, and ${(1-zt)}^a=1$ at $t=0$. … ►In (15.6.6) the integration contour separates the poles of $\mathrm{\Gamma }\left(a+t\right)$ and $\mathrm{\Gamma }\left(b+t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$, and ${(-z)}^t$ has its principal value. … ►In each of (15.6.8) and (15.6.9) all functions in the integrand assume their principal values. …

30: 32.7 Bäcklund Transformations

… ►With the exception of ${\text{P}}_{\text{I}}$, a Bäcklund transformation relates a Painlevé transcendent of one type either to another of the same type but with different values of the parameters, or to another type. …