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1: 23.15 Definitions
If, in addition, f ( τ ) 0 as q 0 , then f ( τ ) is called a cusp form. …
2: 23.18 Modular Transformations
and λ ( τ ) is a cusp form of level zero for the corresponding subgroup of SL ( 2 , ) . …
3: 36.2 Catastrophes and Canonical Integrals
Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension K
4: 36.7 Zeros
Inside the cusp, that is, for x 2 < 8 | y | 3 / 27 , 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 z -axis that is far from the origin, the zero contours form an array of rings close to the planes …Away from the z -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: 36.5 Stokes Sets
K = 2 . 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 z -axis by 2 π / 3 . … 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. …
6: 36.10 Differential Equations
In terms of the normal form (36.2.1) the Ψ K ( x ) satisfy the operator equation … K = 2 , cusp: … K = 2 , cusp: … In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations …
7: Bibliography C
  • B. C. Carlson (1972b) Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).
  • M. A. Chaudhry, N. M. Temme, and E. J. M. Veling (1996) Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.
  • J. N. L. Connor and D. Farrelly (1981) Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives. Chem. Phys. Lett. 81 (2), pp. 306–310.
  • G. Cornell, J. H. Silverman, and G. Stevens (Eds.) (1997) Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.
  • Cunningham Project (website)