real and imaginary parts
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31: 20.10 Integrals
32: 20.1 Special Notation
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the nome, , . Since is not a single-valued function of , it is assumed that is known, even when is specified. Most applications concern the rectangular case , , so that and and are uniquely related. | |
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33: 21.1 Special Notation
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►Lowercase boldface letters or numbers are -dimensional real or complex vectors, either row or column depending on the context.
Uppercase boldface letters are
real or complex matrices.
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►The function is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
positive integers. | |
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( times). | |
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complex, symmetric matrix with strictly positive definite, i.e., a Riemann matrix. | |
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34: 28.20 Definitions and Basic Properties
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►as with , and
…as with .
…as with .
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35: 32.11 Asymptotic Approximations for Real Variables
36: 6.7 Integral Representations
37: 7.7 Integral Representations
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7.7.3
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38: 4.38 Inverse Hyperbolic Functions: Further Properties
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4.38.6
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4.38.7
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►which requires
to lie between the two rectangular hyperbolas given by
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4.38.10
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4.38.12
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39: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►At negative energies and both and are purely imaginary.
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Scaling
►The -scaled variables and of §13.2 are given by … ►Searches for resonances as poles of the -matrix in the complex half-plane . See for example Csótó and Hale (1997).
Eigenstates using complex-rotated coordinates , so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).