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1: 36.2 Catastrophes and Canonical Integrals
§36.2 Catastrophes and Canonical Integrals
§36.2(i) Definitions
Canonical Integrals
§36.2(ii) Special Cases
§36.2(iii) Symmetries
2: 36.3 Visualizations of Canonical Integrals
§36.3 Visualizations of Canonical Integrals
§36.3(i) Canonical Integrals: Modulus
§36.3(ii) Canonical Integrals: Phase
In Figure 36.3.13(a) points of confluence of phase contours are zeros of Ψ 2 ( x , y ) ; similarly for other contour plots in this subsection. In Figure 36.3.13(b) points of confluence of all colors are zeros of Ψ 2 ( x , y ) ; similarly for other density plots in this subsection. …
3: 1.12 Continued Fractions
1.12.4 C n = b 0 + a 1 b 1 + a 2 b 2 + a n b n = A n B n .
A n and B n are called the n th (canonical) numerator and denominator respectively. …
1.12.7 A n B n 1 B n A n 1 = ( 1 ) n 1 k = 1 n a k , n = 0 , 1 , 2 , .
1.12.8 C n C n 1 = ( 1 ) n 1 k = 1 n a k B n 1 B n , n = 1 , 2 , 3 , ,
4: 36.14 Other Physical Applications
§36.14(i) Caustics
§36.14(ii) Optics
§36.14(iii) Quantum Mechanics
§36.14(iv) Acoustics
5: 36.10 Differential Equations
§36.10 Differential Equations
§36.10(i) Equations for Ψ K ( 𝐱 )
In terms of the normal form (36.2.1) the Ψ K ( 𝐱 ) satisfy the operator equation … K = 3 , swallowtail: … In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( 𝐱 ) satisfy the following operator equations …
6: 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 .
7: 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. …
8: 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)
9: 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 ± .
10: 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 ) ) ,