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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: 36.4 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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§36.4(ii) Visualizations
…3: 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). … ►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the -axis by . … ►This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. …4: 36.7 Zeros
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§36.7(ii) Cusp Canonical Integral
… ► … ►Inside the cusp, that is, for , the zeros form pairs lying in curved rows. … ►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 …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: 20 Theta Functions
Chapter 20 Theta Functions
…6: 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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7: 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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8: 23.15 Definitions
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►The set of all bilinear transformations of this form is denoted by SL (Serre (1973, p. 77)).
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►If, in addition, as , then is called a cusp form.
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23.15.8
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23.15.9
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9: 8 Incomplete Gamma and Related
Functions
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10: 28 Mathieu Functions and Hill’s Equation
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