Regge poles

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1: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(vii) Complex Variables and Parameters

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Searches for resonances as poles of the $S$-matrix in the complex half-plane . See for example Csótó and Hale (1997).

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Regge poles at complex values of $\mathrm{\ell }$. See for example Takemasa et al. (1979).

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2: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►In the upper half of the image, the poles of $\mathrm{\Gamma }\left(z\right)$ are clearly visible at negative integer values of $z$: the phase changes by $2\pi$ around each pole, showing a full revolution of the color wheel. … ►In the lower half of the image, the poles of $\psi \left(z\right)$ (corresponding to the poles of $\mathrm{\Gamma }\left(z\right)$) and the zeros between them are clear. Phase changes around the zeros are of opposite sign to those around the poles. …

3: 10.72 Mathematical Applications

… ►The number $m$ can also be replaced by any real constant $\lambda$ $(>-2)$ in the sense that ${(z-z_0)}^{-\lambda }$ $f(z)$ is analytic and nonvanishing at $z_0$; moreover, $g(z)$ is permitted to have a single or double pole at $z_0$. The order of the approximating Bessel functions, or modified Bessel functions, is $1/(\lambda +2)$, except in the case when $g(z)$ has a double pole at $z_0$. … ►

§10.72(ii) Differential Equations with Poles

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§10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point

►In (10.72.1) assume $f(z)=f(z,\alpha )$ and $g(z)=g(z,\alpha )$ depend continuously on a real parameter $\alpha$, $f(z,\alpha )$ has a simple zero $z=z_0(\alpha )$ and a double pole $z=0$, except for a critical value $\alpha =a$, where $z_0(a)=0$. …

4: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►Equations (34.5.9) and (34.5.10) are called Regge symmetries. Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). …

5: 22.4 Periods, Poles, and Zeros

§22.4 Periods, Poles, and Zeros

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§22.4(i) Distribution

►For each Jacobian function, Table 22.4.1 gives its periods in the $z$-plane in the left column, and the position of one of its poles in the second row. … ►The other poles and zeros are at the congruent points. … ►Using the p,q notation of (22.2.10), Figure 22.4.2 serves as a mnemonic for the poles, zeros, periods, and half-periods of the 12 Jacobian elliptic functions as follows. …

6: 4.28 Definitions and Periodicity

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Periodicity and Zeros

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7: 13.27 Mathematical Applications

… ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …

8: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). …

9: Bibliography V

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Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories. Il Nuovo Cimento A 57 (1), pp. 190–197.

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§5.20.

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10: 8.6 Integral Representations

… ►where the integration path passes above or below the pole at $t=1$, according as upper or lower signs are taken. … ►In (8.6.10)–(8.6.12), $c$ is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at $s=a$, in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at $s=0,1,2,\mathrm{\dots }$. …