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1: 3.8 Nonlinear Equations
(More precisely, p is the largest of the possible set of indices for (3.8.3).) … For multiple zeros the convergence is linear, but if the multiplicity m is known then quadratic convergence can be restored by multiplying the ratio f ( z n ) / f ( z n ) in (3.8.4) by m . … It is called a Julia set. In general the Julia set of an analytic function f ( z ) is a fractal, that is, a set that is self-similar. See Julia (1918) and Devaney (1986). …
2: Bibliography J
  • A. Jonquière (1889) Note sur la série n = 1 x n / n s . Bull. Soc. Math. France 17, pp. 142–152 (French).
  • G. Julia (1918) Memoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1), pp. 47–245 (French).
  • 3: 4.30 Elementary Properties
    Table 4.30.1: Hyperbolic functions: interrelations. …
    sinh θ = a cosh θ = a tanh θ = a csch θ = a sech θ = a coth θ = a
    sinh θ a ( a 2 - 1 ) 1 / 2 a ( 1 - a 2 ) - 1 / 2 a - 1 a - 1 ( 1 - a 2 ) 1 / 2 ( a 2 - 1 ) - 1 / 2
    cosh θ ( 1 + a 2 ) 1 / 2 a ( 1 - a 2 ) - 1 / 2 a - 1 ( 1 + a 2 ) 1 / 2 a - 1 a ( a 2 - 1 ) - 1 / 2
    tanh θ a ( 1 + a 2 ) - 1 / 2 a - 1 ( a 2 - 1 ) 1 / 2 a ( 1 + a 2 ) - 1 / 2 ( 1 - a 2 ) 1 / 2 a - 1
    csch θ a - 1 ( a 2 - 1 ) - 1 / 2 a - 1 ( 1 - a 2 ) 1 / 2 a a ( 1 - a 2 ) - 1 / 2 ( a 2 - 1 ) 1 / 2
    sech θ ( 1 + a 2 ) - 1 / 2 a - 1 ( 1 - a 2 ) 1 / 2 a ( 1 + a 2 ) - 1 / 2 a a - 1 ( a 2 - 1 ) 1 / 2
    4: 17.10 Transformations of ψ r r Functions
    17.10.1 ψ 2 2 ( a , b c , d ; q , z ) = ( a z , d / a , c / b , d q / ( a b z ) ; q ) ( z , d , q / b , c d / ( a b z ) ; q ) ψ 2 2 ( a , a b z / d a z , c ; q , d a ) ,
    17.10.3 ψ 8 8 ( q a 1 2 , - q a 1 2 , c , d , e , f , a q - n , q - n a 1 2 , - a 1 2 , a q / c , a q / d , a q / e , a q / f , q n + 1 , a q n + 1 ; q , a 2 q 2 n + 2 c d e f ) = ( a q , q / a , a q / ( c d ) , a q / ( e f ) ; q ) n ( q / c , q / d , a q / e , a q / f ; q ) n ψ 4 4 ( e , f , a q n + 1 / ( c d ) , q - n a q / c , a q / d , q n + 1 , e f / ( a q n ) ; q , q ) ,
    17.10.4 ψ 2 2 ( e , f a q / c , a q / d ; q , a q e f ) = ( q / c , q / d , a q / e , a q / f ; q ) ( a q , q / a , a q / ( c d ) , a q / ( e f ) ; q ) n = - ( 1 - a q 2 n ) ( c , d , e , f ; q ) n ( 1 - a ) ( a q / c , a q / d , a q / e , a q / f ; q ) n ( q a 3 c d e f ) n q n 2 .
    17.10.5 ( a q / b , a q / c , a q / d , a q / e , q / ( a b ) , q / ( a c ) , q / ( a d ) , q / ( a e ) ; q ) ( f a , g a , f / a , g / a , q a 2 , q / a 2 ; q ) ψ 8 8 ( q a , - q a , b a , c a , d a , e a , f a , g a a , - a , a q / b , a q / c , a q / d , a q / e , a q / f , a q / g ; q , q 2 b c d e f g ) = ( q , q / ( b f ) , q / ( c f ) , q / ( d f ) , q / ( e f ) , q f / b , q f / c , q f / d , q f / e ; q ) ( f a , q / ( f a ) , a q / f , f / a , g / f , f g , q f 2 ; q ) ϕ 7 8 ( f 2 , q f , - q f , f b , f c , f d , f e , f g f , - f , f q / b , f q / c , f q / d , f q / e , f q / g ; q , q 2 b c d e f g ) + idem ( f ; g ) .
    17.10.6 ( a q / b , a q / c , a q / d , a q / e , a q / f , q / ( a b ) , q / ( a c ) , q / ( a d ) , q / ( a e ) , q / ( a f ) ; q ) ( a g , a h , a k , g / a , h / a , k / a , q a 2 , q / a 2 ; q ) ψ 10 10 ( q a , - q a , b a , c a , d a , e a , f a , g a , h a , k a a , - a , a q / b , a q / c , a q / d , a q / e , a q / f , a q / g , a q / h , a q / k ; q , q 2 b c d e f g h k ) = ( q , q / ( b g ) , q / ( c g ) , q / ( d g ) , q / ( e g ) , q / ( f g ) , q g / b , q g / c , q g / d , q g / e , q g / f ; q ) ( g h , g k , h / g , a g , q / ( a g ) , g / a , a q / g , q g 2 ; q ) ϕ 9 10 ( g 2 , q g , - q g , g b , g c , g d , g e , g f , g h , g k g , - g , q g / b , q g / c , q g / d , q g / e , q g / f , q g / h , q g / k ; q , q 2 b c d e f g h k ) + idem ( g ; h , k ) .
    5: 17.8 Special Cases of ψ r r Functions
    17.8.4 ψ 2 2 ( b , c ; a q / b , a q / c ; q , - a q / ( b c ) ) = ( a q / ( b c ) ; q ) ( a q 2 / b 2 , a q 2 / c 2 , q 2 , a q , q / a ; q 2 ) ( a q / b , a q / c , q / b , q / c , - a q / ( b c ) ; q ) ,
    17.8.5 ψ 3 3 ( b , c , d q / b , q / c , q / d ; q , q b c d ) = ( q , q / ( b c ) , q / ( b d ) , q / ( c d ) ; q ) ( q / b , q / c , q / d , q / ( b c d ) ; q ) ,
    17.8.6 ψ 4 4 ( - q a 1 2 , b , c , d - a 1 2 , a q / b , a q / c , a q / d ; q , q a 3 2 b c d ) = ( a q , a q / ( b c ) , a q / ( b d ) , a q / ( c d ) , q a 1 2 / b , q a 1 2 / c , q a 1 2 / d , q , q / a ; q ) ( a q / b , a q / c , a q / d , q / b , q / c , q / d , q a 1 2 , q a - 1 2 , q a 3 2 / ( b c d ) ; q ) ,
    17.8.7 ψ 6 6 ( q a 1 2 , - q a 1 2 , b , c , d , e a 1 2 , - a 1 2 , a q / b , a q / c , a q / d , a q / e ; q , q a 2 b c d e ) = ( a q , a q / ( b c ) , a q / ( b d ) , a q / ( b e ) , a q / ( c d ) , a q / ( c e ) , a q / ( d e ) , q , q / a ; q ) ( a q / b , a q / c , a q / d , a q / e , q / b , q / c , q / d , q / e , q a 2 / ( b c d e ) ; q ) .
    17.8.8 ψ 2 2 ( b 2 , b 2 / c q , c q ; q 2 , c q 2 / b 2 ) = 1 2 ( q 2 , q b 2 , q / b 2 , c q / b 2 ; q 2 ) ( c q , c q 2 / b 2 , q 2 / b 2 , c / b 2 ; q 2 ) ( ( c q / b ; q ) ( b q ; q ) + ( - c q / b ; q ) ( - b q ; q ) ) , | c q 2 | < | b 2 | .
    6: 17.9 Further Transformations of ϕ r r + 1 Functions
    17.9.2 ϕ 1 2 ( q - n , b c ; q , z ) = ( c / b ; q ) n ( c ; q ) n b n ϕ 1 3 ( q - n , b , q / z b q 1 - n / c ; q , z / c ) ,
    17.9.12 ϕ 2 3 ( a , b , c d , e ; q , d e a b c ) = ( e / b , e / c , c q / a , q / d ; q ) ( e , c q / d , q / a , e / ( b c ) ; q ) ϕ 2 3 ( c , d / a , c q / e c q / a , b c q / e ; q , b q d ) - ( q / d , e q / d , b , c , d / a , d e / ( b c q ) , b c q 2 / ( d e ) ; q ) ( d / q , e , b q / d , c q / d , q / a , e / ( b c ) , b c q / e ; q ) ϕ 2 3 ( a q / d , b q / d , c q / d q 2 / d , e q / d ; q , d e a b c ) ,
    17.9.13 ϕ 2 3 ( a , b , c d , e ; q , d e a b c ) = ( e / b , e / c ; q ) ( e , e / ( b c ) ; q ) ϕ 2 3 ( d / a , b , c d , b c q / e ; q , q ) + ( d / a , b , c , d e / ( b c ) ; q ) ( d , e , b c / e , d e / ( a b c ) ; q ) ϕ 2 3 ( e / b , e / c , d e / ( a b c ) d e / ( b c ) , e q / ( b c ) ; q , q ) .
    17.9.14 ϕ 3 4 ( q - n , a , b , c d , e , f ; q , q ) = ( e / a , f / a ; q ) n ( e , f ; q ) n a n ϕ 3 4 ( q - n , a , d / b , d / c d , a q 1 - n / e , a q 1 - n / f ; q , q ) = ( a , e f / ( a b ) , e f / ( a c ) ; q ) n ( e , f , e f / ( a b c ) ; q ) n ϕ 3 4 ( q - n , e / a , f / a , e f / ( a b c ) e f / ( a b ) , e f / ( a c ) , q 1 - n / a ; q , q ) .
    17.9.16 ϕ 7 8 ( a , q a 1 2 , - q a 1 2 , b , c , d , e , f a 1 2 , - a 1 2 , a q / b , a q / c , a q / d , a q / e , a q / f ; q , a 2 q 2 b c d e f ) = ( a q , a q / ( d e ) , a q / ( d f ) , a q / ( e f ) ; q ) ( a q / d , a q / e , a q / f , a q / ( d e f ) ; q ) ϕ 3 4 ( a q / ( b c ) , d , e , f a q / b , a q / c , d e f / a ; q , q ) + ( a q , a q / ( b c ) , d , e , f , a 2 q 2 / ( b d e f ) , a 2 q 2 / ( c d e f ) ; q ) ( a q / b , a q / c , a q / d , a q / e , a q / f , a 2 q 2 / ( b c d e f ) , d e f / ( a q ) ; q ) ϕ 3 4 ( a q / ( d e ) , a q / ( d f ) , a q / ( e f ) , a 2 q 2 / ( b c d e f ) a 2 q 2 / ( b d e f ) , a 2 q 2 / ( c d e f ) , a q 2 / ( d e f ) ; q , q ) .
    7: 36.5 Stokes Sets
    §36.5 Stokes Sets
    §36.5(i) Definitions
    §36.5(ii) Cuspoids
    Elliptic Umbilic Stokes Set (Codimension three)
    §36.5(iv) Visualizations
    8: 36.4 Bifurcation Sets
    §36.4 Bifurcation Sets
    Bifurcation (Catastrophe) Set for Cuspoids
    Bifurcation (Catastrophe) Set for Umbilics
    K = 1 , fold bifurcation set: …
    §36.4(ii) Visualizations
    9: 21.1 Special Notation
    g , h

    positive integers.

    g × h

    set of all g × h matrices with integer elements.

    S g

    set of g -dimensional vectors with elements in S .

    | S |

    number of elements of the set S .

    S 1 / S 2

    set of all elements of S 1 , modulo elements of S 2 . Thus two elements of S 1 / S 2 are equivalent if they are both in S 1 and their difference is in S 2 . (For an example see §20.12(ii).)

    The function Θ ( ϕ | B ) = θ ( ϕ / ( 2 π i ) | B / ( 2 π i ) ) is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
    10: 4.16 Elementary Properties
    Table 4.16.3: Trigonometric functions: interrelations. …
    sin θ = a cos θ = a tan θ = a csc θ = a sec θ = a cot θ = a
    sin θ a ( 1 - a 2 ) 1 / 2 a ( 1 + a 2 ) - 1 / 2 a - 1 a - 1 ( a 2 - 1 ) 1 / 2 ( 1 + a 2 ) - 1 / 2
    cos θ ( 1 - a 2 ) 1 / 2 a ( 1 + a 2 ) - 1 / 2 a - 1 ( a 2 - 1 ) 1 / 2 a - 1 a ( 1 + a 2 ) - 1 / 2
    tan θ a ( 1 - a 2 ) - 1 / 2 a - 1 ( 1 - a 2 ) 1 / 2 a ( a 2 - 1 ) - 1 / 2 ( a 2 - 1 ) 1 / 2 a - 1
    csc θ a - 1 ( 1 - a 2 ) - 1 / 2 a - 1 ( 1 + a 2 ) 1 / 2 a a ( a 2 - 1 ) - 1 / 2 ( 1 + a 2 ) 1 / 2
    sec θ ( 1 - a 2 ) - 1 / 2 a - 1 ( 1 + a 2 ) 1 / 2 a ( a 2 - 1 ) - 1 / 2 a a - 1 ( 1 + a 2 ) 1 / 2