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1: 36.2 Catastrophes and Canonical Integrals
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Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension
… ►Special cases: , fold catastrophe; , cusp catastrophe; , swallowtail catastrophe. … ►Canonical Integrals
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36.2.28
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36.2.29
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2: 36.5 Stokes Sets
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. Cusp
►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ► …3: 36.4 Bifurcation Sets
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, cusp bifurcation set:
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►Swallowtail cusp lines (ribs):
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►Elliptic umbilic bifurcation set (codimension three): for fixed , the section of the bifurcation set is a three-cusped astroid
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►Elliptic umbilic cusp lines (ribs):
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►Hyperbolic umbilic cusp line (rib):
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4: 36.7 Zeros
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§36.7(ii) Cusp Canonical Integral
… ► … ►Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the -axis that is far from the origin, the zero contours form an array of rings close to the planes …Away from the -axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral. …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. …5: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water.
J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.
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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface.
J. Acoust. Soc. Amer. 95 (2), pp. 650–660.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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6: Bibliography P
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The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic.
Philos. Mag. (7) 37, pp. 311–317.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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7: 20 Theta Functions
Chapter 20 Theta Functions
…8: Bibliography W
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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The Stokes set of the cusp diffraction catastrophe.
J. Phys. A 13 (9), pp. 2913–2928.
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9: 23.15 Definitions
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►If, in addition, as , then is called a cusp form.
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10: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces.
J. Phys. A 8 (4), pp. 566–584.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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