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1: 36.2 Catastrophes and Canonical Integrals
Canonical Integrals
Ψ ( E ) ( 0 ) = 1 3 π Γ ( 1 6 ) = 3.28868 ,
36.2.24 Ψ ( U ) ( x , y , z ) = Ψ ( U ) ( x , y , - z ) ¯ , U = E , H .
36.2.25 Ψ ( E ) ( x , - y , z ) = Ψ ( E ) ( x , y , z ) .
36.2.27 Ψ ( H ) ( x , y , z ) = Ψ ( H ) ( y , x , z ) .
2: 36.3 Visualizations of Canonical Integrals
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 ) . …
Figure 36.3.18: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 0 ) . …
Figure 36.3.19: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 1 ) . …
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 ,
4: 36.1 Special Notation
§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: 36.9 Integral Identities
§36.9 Integral Identities
36.9.1 | Ψ 1 ( x ) | 2 = 2 5 / 3 0 Ψ 1 ( 2 2 / 3 ( 3 u 2 + x ) ) d u ;
36.9.8 | Ψ ( H ) ( x , y , z ) | 2 = 8 π 2 ( 2 9 ) 1 / 3 - - Ai ( ( 4 3 ) 1 / 3 ( x + z v + 3 u 2 ) ) Ai ( ( 4 3 ) 1 / 3 ( y + z u + 3 v 2 ) ) d u d v .
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 .
6: 36.6 Scaling Relations
§36.6 Scaling Relations
Ψ K ( x ; k ) = k β K Ψ K ( y ( k ) ) ,
Ψ ( U ) ( x ; k ) = k β ( U ) Ψ ( U ) ( y ( U ) ( k ) ) ,
Indices for k -Scaling of Magnitude of Ψ K or Ψ ( U ) (Singularity Index)
7: 36.11 Leading-Order Asymptotics
§36.11 Leading-Order Asymptotics
and far from the bifurcation set, the cuspoid canonical integrals are approximated by …
36.11.5 Ψ 3 ( 0 , y , 0 ) = Ψ 3 ( 0 , - y , 0 ) ¯ = exp ( 1 4 i π ) π / y ( 1 - ( i / 3 ) exp ( 3 2 i ( 2 y / 5 ) 5 / 3 ) + o ( 1 ) ) , y + .
36.11.7 Ψ ( E ) ( 0 , 0 , z ) = π z ( i + 3 exp ( 4 27 i z 3 ) + o ( 1 ) ) , z ± ,
36.11.8 Ψ ( H ) ( 0 , 0 , z ) = 2 π z ( 1 - i 3 exp ( 1 27 i z 3 ) + o ( 1 ) ) , z ± .
8: 36.8 Convergent Series Expansions
§36.8 Convergent Series Expansions
Ψ K ( x ) = 2 K + 2 n = 0 exp ( i π ( 2 n + 1 ) 2 ( K + 2 ) ) Γ ( 2 n + 1 K + 2 ) a 2 n ( x ) , K even,
For multinomial power series for Ψ K ( x ) , see Connor and Curtis (1982).
36.8.3 3 2 / 3 4 π 2 Ψ ( H ) ( 3 1 / 3 x ) = Ai ( x ) Ai ( y ) n = 0 ( - 3 - 1 / 3 i z ) n c n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( - 3 - 1 / 3 i z ) n c n ( x ) d n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( - 3 - 1 / 3 i z ) n d n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 1 ( - 3 - 1 / 3 i z ) n d n ( x ) d n ( y ) n ! ,
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 ) ) ,
9: 36.14 Other Physical Applications
§36.14(i) Caustics
§36.14(ii) Optics
§36.14(iii) Quantum Mechanics
§36.14(iv) Acoustics
10: 36.7 Zeros
§36.7 Zeros
§36.7(i) Fold Canonical Integral
§36.7(ii) Cusp 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 …