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elliptic umbilic canonical integral

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1: 36.2 Catastrophes and Canonical Integrals
Canonical Integrals
Ψ ( E ) ( 0 ) = 1 3 π Γ ( 1 6 ) = 3.28868 ,
36.2.25 Ψ ( E ) ( x , - y , z ) = Ψ ( E ) ( x , y , z ) .
36.2.26 Ψ ( E ) ( - 1 2 x 3 2 y , ± 3 2 x - 1 2 y , z ) = Ψ ( E ) ( x , y , z ) ,
36.2.28 Ψ ( E ) ( 0 , 0 , z ) = Ψ ( E ) ( 0 , 0 , - z ) ¯ = 2 π π z 27 exp ( 2 27 i z 3 ) ( J - 1 / 6 ( 2 27 z 3 ) + i J 1 / 6 ( 2 27 z 3 ) ) , z 0 ,
2: 36.3 Visualizations of Canonical Integrals
Figure 36.3.6: Modulus of elliptic umbilic canonical integral function | Ψ ( E ) ( x , y , 0 ) | . …
Figure 36.3.7: Modulus of elliptic umbilic canonical integral function | Ψ ( E ) ( x , y , 2 ) | . …
Figure 36.3.15: Phase of elliptic umbilic canonical integral ph Ψ ( E ) ( x , y , 0 ) . …
Figure 36.3.16: Phase of elliptic umbilic canonical integral ph Ψ ( E ) ( x , y , 2 ) . …
Figure 36.3.17: Phase of elliptic umbilic canonical integral ph Ψ ( E ) ( x , y , 4 ) . …
3: 36.10 Differential Equations
In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations …
36.10.13 6 2 Ψ ( E ) x y - 2 i z Ψ ( E ) y + y Ψ ( E ) = 0 ,
36.10.14 3 ( 2 Ψ ( E ) x 2 - 2 Ψ ( E ) y 2 ) + 2 i z Ψ ( E ) x - x Ψ ( E ) = 0 .
4: 36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes Φ K ( t ; x ) ; umbilic catastrophes with codimension three Φ ( E ) ( s , t ; x ) , Φ ( H ) ( s , t ; x ) ; canonical integrals Ψ K ( x ) , Ψ ( E ) ( x ) , Ψ ( H ) ( x ) ; diffraction catastrophes Ψ K ( x ; k ) , Ψ ( E ) ( x ; k ) , Ψ ( H ) ( x ; k ) generated by the catastrophes. …
5: 36.8 Convergent Series Expansions
§36.8 Convergent Series Expansions
36.8.4 Ψ ( E ) ( x ) = 2 π 2 ( 2 3 ) 2 / 3 n = 0 ( - i ( 2 / 3 ) 2 / 3 z ) n n ! ( f n ( x + i y 12 1 / 3 , x - i y 12 1 / 3 ) ) ,
6: 36.9 Integral Identities
36.9.9 | Ψ ( E ) ( x , y , z ) | 2 = 8 π 2 3 2 / 3 0 0 2 π ( Ai ( 1 3 1 / 3 ( x + i y + 2 z u exp ( i θ ) + 3 u 2 exp ( - 2 i θ ) ) ) Bi ( 1 3 1 / 3 ( x - i y + 2 z u exp ( - i θ ) + 3 u 2 exp ( 2 i θ ) ) ) ) u d u d θ .
7: 36.11 Leading-Order Asymptotics
§36.11 Leading-Order Asymptotics
36.11.7 Ψ ( E ) ( 0 , 0 , z ) = π z ( i + 3 exp ( 4 27 i z 3 ) + o ( 1 ) ) , z ± ,
8: 36.7 Zeros
§36.7(iii) Elliptic Umbilic Canonical Integral
The zeros are lines in x = ( x , y , z ) space where ph Ψ ( E ) ( x ) is undetermined. …Near z = z n , and for small x and y , the modulus | Ψ ( E ) ( x ) | has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose z and x repeat distances are given by …
9: Errata
  • Equation (36.10.14)

    36.10.14
    3 ( 2 Ψ ( E ) x 2 - 2 Ψ ( E ) y 2 ) + 2 i z Ψ ( E ) x - x Ψ ( E ) = 0

    Originally this equation appeared with Ψ ( H ) x in the second term, rather than Ψ ( E ) x .

    Reported 2010-04-02.

  • 10: 36.6 Scaling Relations
    §36.6 Scaling Relations